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The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation | What's new
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href="https://terrytao.wordpress.com/tag/elementary-number-theory/" rel="tag">elementary number theory</a>, <a href="https://terrytao.wordpress.com/tag/euler-summation-formula/" rel="tag">Euler summation formula</a>, <a href="https://terrytao.wordpress.com/tag/riemann-zeta-function/" rel="tag">Riemann zeta function</a> | by <a href="https://terrytao.wordpress.com/author/teorth/" title="Posts by Terence Tao" rel="author">Terence Tao</a> </p> </div> <div class="post-content"> <p>The <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann zeta function</a> <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(s)}" class="latex" /> is defined in the region <img src="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29%3E1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29%3E1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29%3E1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\hbox{Re}(s)>1}" class="latex" /> by the absolutely convergent series <a name="zeta-def"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29+%3D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%3D+1+%2B+%5Cfrac%7B1%7D%7B2%5Es%7D+%2B+%5Cfrac%7B1%7D%7B3%5Es%7D+%2B+%5Cldots.+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29+%3D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%3D+1+%2B+%5Cfrac%7B1%7D%7B2%5Es%7D+%2B+%5Cfrac%7B1%7D%7B3%5Es%7D+%2B+%5Cldots.+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29+%3D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%3D+1+%2B+%5Cfrac%7B1%7D%7B2%5Es%7D+%2B+%5Cfrac%7B1%7D%7B3%5Es%7D+%2B+%5Cldots.+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \ldots. \ \ \ \ \ (1)" class="latex" /></p> <p><a name="zeta-def"></a> Thus, for instance, it is known that <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%282%29%3D%5Cpi%5E2%2F6%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%282%29%3D%5Cpi%5E2%2F6%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%282%29%3D%5Cpi%5E2%2F6%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(2)=\pi^2/6}" class="latex" />, and thus <a name="zeta+2"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+1+%2B+%5Cfrac%7B1%7D%7B4%7D+%2B+%5Cfrac%7B1%7D%7B9%7D+%2B+%5Cldots+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D.+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+1+%2B+%5Cfrac%7B1%7D%7B4%7D+%2B+%5Cfrac%7B1%7D%7B9%7D+%2B+%5Cldots+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D.+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+1+%2B+%5Cfrac%7B1%7D%7B4%7D+%2B+%5Cfrac%7B1%7D%7B9%7D+%2B+%5Cldots+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D.+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \ldots = \frac{\pi^2}{6}. \ \ \ \ \ (2)" class="latex" /></p> <p><a name="zeta+2"></a></p> <p>For <img src="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\hbox{Re}(s) \leq 1}" class="latex" />, the series on the right-hand side of <a href="#zeta-def">(1)</a> is no longer absolutely convergent, or even conditionally convergent. Nevertheless, the <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta}" class="latex" /> function can be extended to this region (with a pole at <img src="https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s=1}" class="latex" />) by <a href="http://en.wikipedia.org/wiki/Analytic_continuation">analytic continuation</a>. For instance, it can be shown that after analytic continuation, one has <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%280%29+%3D+-1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%280%29+%3D+-1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%280%29+%3D+-1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(0) = -1/2}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28-1%29+%3D+-1%2F12%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28-1%29+%3D+-1%2F12%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28-1%29+%3D+-1%2F12%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(-1) = -1/12}" class="latex" />, and <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28-2%29%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28-2%29%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28-2%29%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(-2)=0}" class="latex" />, and more generally <a name="zeta-bern"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28-s%29+%3D+-+%5Cfrac%7BB_%7Bs%2B1%7D%7D%7Bs%2B1%7D+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28-s%29+%3D+-+%5Cfrac%7BB_%7Bs%2B1%7D%7D%7Bs%2B1%7D+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28-s%29+%3D+-+%5Cfrac%7BB_%7Bs%2B1%7D%7D%7Bs%2B1%7D+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \zeta(-s) = - \frac{B_{s+1}}{s+1} \ \ \ \ \ (3)" class="latex" /></p> <p><a name="zeta-bern"></a> for <img src="https://s0.wp.com/latex.php?latex=%7Bs%3D1%2C2%2C%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%3D1%2C2%2C%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%3D1%2C2%2C%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s=1,2,\ldots}" class="latex" />, where <img src="https://s0.wp.com/latex.php?latex=%7BB_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B_n}" class="latex" /> are the <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli numbers</a>. If one <em>formally</em> applies <a href="#zeta-def">(1)</a> at these values of <img src="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s}" class="latex" />, one obtains the somewhat bizarre formulae <a name="zeta-1"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+1+%3D+1+%2B+1+%2B+1+%2B+%5Cldots+%3D+-1%2F2+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+1+%3D+1+%2B+1+%2B+1+%2B+%5Cldots+%3D+-1%2F2+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+1+%3D+1+%2B+1+%2B+1+%2B+%5Cldots+%3D+-1%2F2+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty 1 = 1 + 1 + 1 + \ldots = -1/2 \ \ \ \ \ (4)" class="latex" /></p> <p><a name="zeta-1"></a> <a name="zeta-2"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n+%3D+1+%2B+2+%2B+3+%2B+%5Cldots+%3D+-1%2F12+%5C+%5C+%5C+%5C+%5C+%285%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n+%3D+1+%2B+2+%2B+3+%2B+%5Cldots+%3D+-1%2F12+%5C+%5C+%5C+%5C+%5C+%285%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n+%3D+1+%2B+2+%2B+3+%2B+%5Cldots+%3D+-1%2F12+%5C+%5C+%5C+%5C+%5C+%285%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty n = 1 + 2 + 3 + \ldots = -1/12 \ \ \ \ \ (5)" class="latex" /></p> <p><a name="zeta-2"></a> <a name="zeta-3"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n%5E2+%3D+1+%2B+4+%2B+9+%2B+%5Cldots+%3D+0+%5C+%5C+%5C+%5C+%5C+%286%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n%5E2+%3D+1+%2B+4+%2B+9+%2B+%5Cldots+%3D+0+%5C+%5C+%5C+%5C+%5C+%286%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n%5E2+%3D+1+%2B+4+%2B+9+%2B+%5Cldots+%3D+0+%5C+%5C+%5C+%5C+%5C+%286%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty n^2 = 1 + 4 + 9 + \ldots = 0 \ \ \ \ \ (6)" class="latex" /></p> <p><a name="zeta-3"></a> and <a name="zeta-s"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n%5Es+%3D+1+%2B+2%5Es+%2B+3%5Es+%2B+%5Cldots+%3D+-%5Cfrac%7BB_%7Bs%2B1%7D%7D%7Bs%2B1%7D.+%5C+%5C+%5C+%5C+%5C+%287%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n%5Es+%3D+1+%2B+2%5Es+%2B+3%5Es+%2B+%5Cldots+%3D+-%5Cfrac%7BB_%7Bs%2B1%7D%7D%7Bs%2B1%7D.+%5C+%5C+%5C+%5C+%5C+%287%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n%5Es+%3D+1+%2B+2%5Es+%2B+3%5Es+%2B+%5Cldots+%3D+-%5Cfrac%7BB_%7Bs%2B1%7D%7D%7Bs%2B1%7D.+%5C+%5C+%5C+%5C+%5C+%287%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty n^s = 1 + 2^s + 3^s + \ldots = -\frac{B_{s+1}}{s+1}. \ \ \ \ \ (7)" class="latex" /></p> <p><a name="zeta-s"></a></p> <p>Clearly, these formulae do not make sense if one stays within the traditional way to evaluate infinite series, and so it seems that one is forced to use the somewhat unintuitive analytic continuation interpretation of such sums to make these formulae rigorous. But as it stands, the formulae look “wrong” for several reasons. Most obviously, the summands on the left are all positive, but the right-hand sides can be zero or negative. A little more subtly, the identities do not appear to be consistent with each other. For instance, if one adds <a href="#zeta-1">(4)</a> to <a href="#zeta-2">(5)</a>, one obtains <a name="sam1"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28n%2B1%29+%3D+2+%2B+3+%2B+4+%2B+%5Cldots+%3D+-7%2F12+%5C+%5C+%5C+%5C+%5C+%288%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28n%2B1%29+%3D+2+%2B+3+%2B+4+%2B+%5Cldots+%3D+-7%2F12+%5C+%5C+%5C+%5C+%5C+%288%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28n%2B1%29+%3D+2+%2B+3+%2B+4+%2B+%5Cldots+%3D+-7%2F12+%5C+%5C+%5C+%5C+%5C+%288%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty (n+1) = 2 + 3 + 4 + \ldots = -7/12 \ \ \ \ \ (8)" class="latex" /></p> <p><a name="sam1"></a> whereas if one subtracts <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" /> from <a href="#zeta-2">(5)</a> one obtains instead <a name="sam2"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D2%7D%5E%5Cinfty+n+%3D+0+%2B+2+%2B+3+%2B+4+%2B+%5Cldots+%3D+-13%2F12+%5C+%5C+%5C+%5C+%5C+%289%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D2%7D%5E%5Cinfty+n+%3D+0+%2B+2+%2B+3+%2B+4+%2B+%5Cldots+%3D+-13%2F12+%5C+%5C+%5C+%5C+%5C+%289%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D2%7D%5E%5Cinfty+n+%3D+0+%2B+2+%2B+3+%2B+4+%2B+%5Cldots+%3D+-13%2F12+%5C+%5C+%5C+%5C+%5C+%289%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=2}^\infty n = 0 + 2 + 3 + 4 + \ldots = -13/12 \ \ \ \ \ (9)" class="latex" /></p> <p><a name="sam2"></a> and the two equations seem inconsistent with each other.</p> <p>However, it is possible to interpret <a href="#zeta-1">(4)</a>, <a href="#zeta-2">(5)</a>, <a href="#zeta-3">(6)</a> by purely real-variable methods, without recourse to complex analysis methods such as analytic continuation, thus giving an “elementary” interpretation of these sums that only requires undergraduate calculus; we will later also explain how this interpretation deals with the apparent inconsistencies pointed out above.</p> <p>To see this, let us first consider a convergent sum such as <a href="#zeta+2">(2)</a>. The classical interpretation of this formula is the assertion that the partial sums</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+1+%2B+%5Cfrac%7B1%7D%7B4%7D+%2B+%5Cfrac%7B1%7D%7B9%7D+%2B+%5Cldots+%2B+%5Cfrac%7B1%7D%7BN%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+1+%2B+%5Cfrac%7B1%7D%7B4%7D+%2B+%5Cfrac%7B1%7D%7B9%7D+%2B+%5Cldots+%2B+%5Cfrac%7B1%7D%7BN%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+1+%2B+%5Cfrac%7B1%7D%7B4%7D+%2B+%5Cfrac%7B1%7D%7B9%7D+%2B+%5Cldots+%2B+%5Cfrac%7B1%7D%7BN%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^N \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \ldots + \frac{1}{N^2}" class="latex" /></p> <p>converge to <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%5E2%2F6%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%5E2%2F6%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%5E2%2F6%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi^2/6}" class="latex" /> as <img src="https://s0.wp.com/latex.php?latex=%7BN+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N \rightarrow \infty}" class="latex" />, or in other words that</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+%2B+o%281%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+%2B+o%281%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+%2B+o%281%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^N \frac{1}{n^2} = \frac{\pi^2}{6} + o(1)" class="latex" /></p> <p>where <img src="https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{o(1)}" class="latex" /> denotes a quantity that goes to zero as <img src="https://s0.wp.com/latex.php?latex=%7BN+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N \rightarrow \infty}" class="latex" />. Actually, by using the <a href="http://en.wikipedia.org/wiki/Integral_test_for_convergence">integral test</a> estimate</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3DN%2B1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D+%5Cleq+%5Cint_N%5E%5Cinfty+%5Cfrac%7Bdx%7D%7Bx%5E2%7D+%3D+%5Cfrac%7B1%7D%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3DN%2B1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D+%5Cleq+%5Cint_N%5E%5Cinfty+%5Cfrac%7Bdx%7D%7Bx%5E2%7D+%3D+%5Cfrac%7B1%7D%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3DN%2B1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D+%5Cleq+%5Cint_N%5E%5Cinfty+%5Cfrac%7Bdx%7D%7Bx%5E2%7D+%3D+%5Cfrac%7B1%7D%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=N+1}^\infty \frac{1}{n^2} \leq \int_N^\infty \frac{dx}{x^2} = \frac{1}{N}" class="latex" /></p> <p>we have the sharper result</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^N \frac{1}{n^2} = \frac{\pi^2}{6} + O(\frac{1}{N})." class="latex" /></p> <p>Thus we can view <img src="https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\frac{\pi^2}{6}}" class="latex" /> as the leading coefficient of the asymptotic expansion of the partial sums of <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+1%2Fn%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+1%2Fn%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+1%2Fn%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{n=1}^\infty 1/n^2}" class="latex" />.</p> <p>One can then try to inspect the partial sums of the expressions in <a href="#zeta-1">(4)</a>, <a href="#zeta-2">(5)</a>, <a href="#zeta-3">(6)</a>, but the coefficients bear no obvious relationship to the right-hand sides:</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+1+%3D+N+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+1+%3D+N+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+1+%3D+N+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^N 1 = N " class="latex" /></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+n+%3D+%5Cfrac%7B1%7D%7B2%7D+N%5E2+%2B+%5Cfrac%7B1%7D%7B2%7D+N&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+n+%3D+%5Cfrac%7B1%7D%7B2%7D+N%5E2+%2B+%5Cfrac%7B1%7D%7B2%7D+N&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+n+%3D+%5Cfrac%7B1%7D%7B2%7D+N%5E2+%2B+%5Cfrac%7B1%7D%7B2%7D+N&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^N n = \frac{1}{2} N^2 + \frac{1}{2} N" class="latex" /></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+n%5E2+%3D+%5Cfrac%7B1%7D%7B3%7D+N%5E3+%2B+%5Cfrac%7B1%7D%7B2%7D+N%5E2+%2B+%5Cfrac%7B1%7D%7B6%7D+N.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+n%5E2+%3D+%5Cfrac%7B1%7D%7B3%7D+N%5E3+%2B+%5Cfrac%7B1%7D%7B2%7D+N%5E2+%2B+%5Cfrac%7B1%7D%7B6%7D+N.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+n%5E2+%3D+%5Cfrac%7B1%7D%7B3%7D+N%5E3+%2B+%5Cfrac%7B1%7D%7B2%7D+N%5E2+%2B+%5Cfrac%7B1%7D%7B6%7D+N.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^N n^2 = \frac{1}{3} N^3 + \frac{1}{2} N^2 + \frac{1}{6} N." class="latex" /></p> <p>For <a href="#zeta-s">(7)</a>, the classical <a href="http://en.wikipedia.org/wiki/Faulhaber%27s_formula">Faulhaber formula</a> (or <em>Bernoulli formula</em>) gives <a name="faul"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+n%5Es+%3D+%5Cfrac%7B1%7D%7Bs%2B1%7D+%5Csum_%7Bj%3D0%7D%5Es+%5Cbinom%7Bs%2B1%7D%7Bj%7D+B_j+N%5E%7Bs%2B1-j%7D+%5C+%5C+%5C+%5C+%5C+%2810%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+n%5Es+%3D+%5Cfrac%7B1%7D%7Bs%2B1%7D+%5Csum_%7Bj%3D0%7D%5Es+%5Cbinom%7Bs%2B1%7D%7Bj%7D+B_j+N%5E%7Bs%2B1-j%7D+%5C+%5C+%5C+%5C+%5C+%2810%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+n%5Es+%3D+%5Cfrac%7B1%7D%7Bs%2B1%7D+%5Csum_%7Bj%3D0%7D%5Es+%5Cbinom%7Bs%2B1%7D%7Bj%7D+B_j+N%5E%7Bs%2B1-j%7D+%5C+%5C+%5C+%5C+%5C+%2810%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^N n^s = \frac{1}{s+1} \sum_{j=0}^s \binom{s+1}{j} B_j N^{s+1-j} \ \ \ \ \ (10)" class="latex" /></p> <p><a name="faul"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+%5Cfrac%7B1%7D%7Bs%2B1%7D+N%5E%7Bs%2B1%7D+%2B+%5Cfrac%7B1%7D%7B2%7D+N%5Es+%2B+%5Cfrac%7Bs%7D%7B12%7D+N%5E%7Bs-1%7D+%2B+%5Cldots+%2B+B_s+N&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+%5Cfrac%7B1%7D%7Bs%2B1%7D+N%5E%7Bs%2B1%7D+%2B+%5Cfrac%7B1%7D%7B2%7D+N%5Es+%2B+%5Cfrac%7Bs%7D%7B12%7D+N%5E%7Bs-1%7D+%2B+%5Cldots+%2B+B_s+N&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+%5Cfrac%7B1%7D%7Bs%2B1%7D+N%5E%7Bs%2B1%7D+%2B+%5Cfrac%7B1%7D%7B2%7D+N%5Es+%2B+%5Cfrac%7Bs%7D%7B12%7D+N%5E%7Bs-1%7D+%2B+%5Cldots+%2B+B_s+N&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle = \frac{1}{s+1} N^{s+1} + \frac{1}{2} N^s + \frac{s}{12} N^{s-1} + \ldots + B_s N" class="latex" /></p> <p>for <img src="https://s0.wp.com/latex.php?latex=%7Bs+%5Cgeq+2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs+%5Cgeq+2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs+%5Cgeq+2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s \geq 2}" class="latex" />, which has a vague resemblance to <a href="#zeta-s">(7)</a>, but again the connection is not particularly clear.</p> <p>The problem here is the discrete nature of the partial sum</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+n%5Es+%3D+%5Csum_%7Bn+%5Cleq+N%7D+n%5Es%2C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+n%5Es+%3D+%5Csum_%7Bn+%5Cleq+N%7D+n%5Es%2C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+n%5Es+%3D+%5Csum_%7Bn+%5Cleq+N%7D+n%5Es%2C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^N n^s = \sum_{n \leq N} n^s," class="latex" /></p> <p>which (if <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" /> is viewed as a real number) has jump discontinuities at each positive integer value of <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" />. These discontinuities yield various artefacts when trying to approximate this sum by a polynomial in <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" />. (These artefacts also occur in <a href="#zeta+2">(2)</a>, but happen in that case to be obscured in the error term <img src="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(1/N)}" class="latex" />; but for the divergent sums <a href="#zeta-1">(4)</a>, <a href="#zeta-2">(5)</a>, <a href="#zeta-3">(6)</a>, <a href="#zeta-s">(7)</a>, they are large enough to cause real trouble.)</p> <p>However, these issues can be resolved by replacing the abruptly truncated partial sums <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5EN+n%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5EN+n%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5EN+n%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{n=1}^N n^s}" class="latex" /> with <a href="http://www.tricki.org/article/Smoothing_sums">smoothed sums</a> <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+n%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+n%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+n%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{n=1}^\infty \eta(n/N) n^s}" class="latex" />, where <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%3A+%7B%5Cbf+R%7D%5E%2B+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%3A+%7B%5Cbf+R%7D%5E%2B+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%3A+%7B%5Cbf+R%7D%5E%2B+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta: {\bf R}^+ \rightarrow {\bf R}}" class="latex" /> is a <em>cutoff function</em>, or more precisely a compactly supported bounded function that is right-continuous the origin and equals <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" /> at <img src="https://s0.wp.com/latex.php?latex=%7B0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{0}" class="latex" />. The case when <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta}" class="latex" /> is the indicator function <img src="https://s0.wp.com/latex.php?latex=%7B1_%7B%5B0%2C1%5D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1_%7B%5B0%2C1%5D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1_%7B%5B0%2C1%5D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1_{[0,1]}}" class="latex" /> then corresponds to the traditional partial sums, with all the attendant discretisation artefacts; but if one chooses a smoother cutoff, then these artefacts begin to disappear (or at least become lower order), and the true asymptotic expansion becomes more manifest.</p> <p>Note that smoothing does not affect the asymptotic value of sums that were already absolutely convergent, thanks to the <a href="http://en.wikipedia.org/wiki/Dominated_convergence_theorem">dominated convergence theorem</a>. For instance, we have</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+%2B+o%281%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+%2B+o%281%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+%2B+o%281%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \eta(n/N) \frac{1}{n^2} = \frac{\pi^2}{6} + o(1)" class="latex" /></p> <p>whenever <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta}" class="latex" /> is a cutoff function (since <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%28n%2FN%29+%5Crightarrow+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%28n%2FN%29+%5Crightarrow+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%28n%2FN%29+%5Crightarrow+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta(n/N) \rightarrow 1}" class="latex" /> pointwise as <img src="https://s0.wp.com/latex.php?latex=%7BN+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N \rightarrow \infty}" class="latex" /> and is uniformly bounded). If <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta}" class="latex" /> is equal to <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" /> on a neighbourhood of the origin, then the integral test argument then recovers the <img src="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(1/N)}" class="latex" /> decay rate:</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \eta(n/N) \frac{1}{n^2} = \frac{\pi^2}{6} + O(\frac{1}{N})." class="latex" /></p> <p>However, smoothing can greatly improve the convergence properties of a divergent sum. The simplest example is <a href="http://en.wikipedia.org/wiki/Grandi%27s_series">Grandi’s series</a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28-1%29%5E%7Bn-1%7D+%3D+1+-+1+%2B+1+-+%5Cldots.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28-1%29%5E%7Bn-1%7D+%3D+1+-+1+%2B+1+-+%5Cldots.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28-1%29%5E%7Bn-1%7D+%3D+1+-+1+%2B+1+-+%5Cldots.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty (-1)^{n-1} = 1 - 1 + 1 - \ldots." class="latex" /></p> <p>The partial sums</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+%28-1%29%5E%7Bn-1%7D+%3D+%5Cfrac%7B1%7D%7B2%7D+%2B+%5Cfrac%7B1%7D%7B2%7D+%28-1%29%5E%7BN-1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+%28-1%29%5E%7Bn-1%7D+%3D+%5Cfrac%7B1%7D%7B2%7D+%2B+%5Cfrac%7B1%7D%7B2%7D+%28-1%29%5E%7BN-1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5EN+%28-1%29%5E%7Bn-1%7D+%3D+%5Cfrac%7B1%7D%7B2%7D+%2B+%5Cfrac%7B1%7D%7B2%7D+%28-1%29%5E%7BN-1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^N (-1)^{n-1} = \frac{1}{2} + \frac{1}{2} (-1)^{N-1}" class="latex" /></p> <p>oscillate between <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7B0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{0}" class="latex" />, and so this series is not conditionally convergent (and certainly not absolutely convergent). However, if one performs analytic continuation on the series</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B%28-1%29%5E%7Bn-1%7D%7D%7Bn%5Es%7D+%3D+1+-+%5Cfrac%7B1%7D%7B2%5Es%7D+%2B+%5Cfrac%7B1%7D%7B3%5Es%7D+-+%5Cldots&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B%28-1%29%5E%7Bn-1%7D%7D%7Bn%5Es%7D+%3D+1+-+%5Cfrac%7B1%7D%7B2%5Es%7D+%2B+%5Cfrac%7B1%7D%7B3%5Es%7D+-+%5Cldots&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B%28-1%29%5E%7Bn-1%7D%7D%7Bn%5Es%7D+%3D+1+-+%5Cfrac%7B1%7D%7B2%5Es%7D+%2B+%5Cfrac%7B1%7D%7B3%5Es%7D+-+%5Cldots&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s} = 1 - \frac{1}{2^s} + \frac{1}{3^s} - \ldots" class="latex" /></p> <p>and sets <img src="https://s0.wp.com/latex.php?latex=%7Bs+%3D+0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs+%3D+0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs+%3D+0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s = 0}" class="latex" />, one obtains a formal value of <img src="https://s0.wp.com/latex.php?latex=%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1/2}" class="latex" /> for this series. This value can also be obtained by smooth summation. Indeed, for any cutoff function <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta}" class="latex" />, we can regroup</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28-1%29%5E%7Bn-1%7D+%5Ceta%28n%2FN%29+%3D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28-1%29%5E%7Bn-1%7D+%5Ceta%28n%2FN%29+%3D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28-1%29%5E%7Bn-1%7D+%5Ceta%28n%2FN%29+%3D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty (-1)^{n-1} \eta(n/N) = " class="latex" /></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B%5Ceta%281%2FN%29%7D%7B2%7D+%2B+%5Csum_%7Bm%3D1%7D%5E%5Cinfty+%5Cfrac%7B%5Ceta%28%282m-1%29%2FN%29+-+2%5Ceta%282m%2FN%29+%2B+%5Ceta%28%282m%2B1%29%2FN%29%7D%7B2%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B%5Ceta%281%2FN%29%7D%7B2%7D+%2B+%5Csum_%7Bm%3D1%7D%5E%5Cinfty+%5Cfrac%7B%5Ceta%28%282m-1%29%2FN%29+-+2%5Ceta%282m%2FN%29+%2B+%5Ceta%28%282m%2B1%29%2FN%29%7D%7B2%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B%5Ceta%281%2FN%29%7D%7B2%7D+%2B+%5Csum_%7Bm%3D1%7D%5E%5Cinfty+%5Cfrac%7B%5Ceta%28%282m-1%29%2FN%29+-+2%5Ceta%282m%2FN%29+%2B+%5Ceta%28%282m%2B1%29%2FN%29%7D%7B2%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{\eta(1/N)}{2} + \sum_{m=1}^\infty \frac{\eta((2m-1)/N) - 2\eta(2m/N) + \eta((2m+1)/N)}{2}." class="latex" /></p> <p>If <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta}" class="latex" /> is twice continuously differentiable (i.e. <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta+%5Cin+C%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta+%5Cin+C%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta+%5Cin+C%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta \in C^2}" class="latex" />), then from Taylor expansion we see that the summand has size <img src="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(1/N^2)}" class="latex" />, and also (from the compact support of <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta}" class="latex" />) is only non-zero when <img src="https://s0.wp.com/latex.php?latex=%7Bm%3DO%28N%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%3DO%28N%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%3DO%28N%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m=O(N)}" class="latex" />. This leads to the asymptotic</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28-1%29%5E%7Bn-1%7D+%5Ceta%28n%2FN%29+%3D+%5Cfrac%7B1%7D%7B2%7D+%2B+O%28+%5Cfrac%7B1%7D%7BN%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28-1%29%5E%7Bn-1%7D+%5Ceta%28n%2FN%29+%3D+%5Cfrac%7B1%7D%7B2%7D+%2B+O%28+%5Cfrac%7B1%7D%7BN%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28-1%29%5E%7Bn-1%7D+%5Ceta%28n%2FN%29+%3D+%5Cfrac%7B1%7D%7B2%7D+%2B+O%28+%5Cfrac%7B1%7D%7BN%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty (-1)^{n-1} \eta(n/N) = \frac{1}{2} + O( \frac{1}{N} )" class="latex" /></p> <p>and so we recover the value of <img src="https://s0.wp.com/latex.php?latex=%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1/2}" class="latex" /> as the leading term of the asymptotic expansion.</p> <blockquote><p><b>Exercise 1</b> Show that if <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta}" class="latex" /> is merely once continuously differentiable (i.e. <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta+%5Cin+C%5E1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta+%5Cin+C%5E1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta+%5Cin+C%5E1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta \in C^1}" class="latex" />), then we have a similar asymptotic, but with an error term of <img src="https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{o(1)}" class="latex" /> instead of <img src="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(1/N)}" class="latex" />. This is an instance of a more general principle that smoother cutoffs lead to better error terms, though the improvement sometimes stops after some degree of regularity.</p></blockquote> <blockquote><p><b>Remark 2</b> The most famous instance of smoothed summation is <a href="http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation">Cesáro summation</a>, which corresponds to the cutoff function <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%28x%29+%3A%3D+%281-x%29_%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%28x%29+%3A%3D+%281-x%29_%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%28x%29+%3A%3D+%281-x%29_%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta(x) := (1-x)_+}" class="latex" />. Unsurprisingly, when Cesáro summation is applied to Grandi’s series, one again recovers the value of <img src="https://s0.wp.com/latex.php?latex=%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1/2}" class="latex" />.</p></blockquote> <p>If we now revisit the divergent series <a href="#zeta-1">(4)</a>, <a href="#zeta-2">(5)</a>, <a href="#zeta-3">(6)</a>, <a href="#zeta-s">(7)</a> with smooth summation in mind, we finally begin to see the origin of the right-hand sides. Indeed, for any fixed smooth cutoff function <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta}" class="latex" />, we will shortly show that <a name="zeta-asym-1"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7B1%7D%7B2%7D+%2B+C_%7B%5Ceta%2C0%7D+N+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2811%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7B1%7D%7B2%7D+%2B+C_%7B%5Ceta%2C0%7D+N+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2811%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7B1%7D%7B2%7D+%2B+C_%7B%5Ceta%2C0%7D+N+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2811%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \eta(n/N) = -\frac{1}{2} + C_{\eta,0} N + O(\frac{1}{N}) \ \ \ \ \ (11)" class="latex" /></p> <p><a name="zeta-asym-1"></a> <a name="zeta-asym-2"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7B1%7D%7B12%7D+%2B+C_%7B%5Ceta%2C1%7D+N%5E2+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2812%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7B1%7D%7B12%7D+%2B+C_%7B%5Ceta%2C1%7D+N%5E2+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2812%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7B1%7D%7B12%7D+%2B+C_%7B%5Ceta%2C1%7D+N%5E2+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2812%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty n \eta(n/N) = -\frac{1}{12} + C_{\eta,1} N^2 + O(\frac{1}{N}) \ \ \ \ \ (12)" class="latex" /></p> <p><a name="zeta-asym-2"></a> <a name="zeta-asym-3"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n%5E2+%5Ceta%28n%2FN%29+%3D+C_%7B%5Ceta%2C2%7D+N%5E3+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2813%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n%5E2+%5Ceta%28n%2FN%29+%3D+C_%7B%5Ceta%2C2%7D+N%5E3+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2813%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n%5E2+%5Ceta%28n%2FN%29+%3D+C_%7B%5Ceta%2C2%7D+N%5E3+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2813%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty n^2 \eta(n/N) = C_{\eta,2} N^3 + O(\frac{1}{N}) \ \ \ \ \ (13)" class="latex" /></p> <p><a name="zeta-asym-3"></a> and more generally <a name="zeta-asym-s"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n%5Es+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7BB_%7Bs%2B1%7D%7D%7Bs%2B1%7D+%2B+C_%7B%5Ceta%2Cs%7D+N%5E%7Bs%2B1%7D+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2814%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n%5Es+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7BB_%7Bs%2B1%7D%7D%7Bs%2B1%7D+%2B+C_%7B%5Ceta%2Cs%7D+N%5E%7Bs%2B1%7D+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2814%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n%5Es+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7BB_%7Bs%2B1%7D%7D%7Bs%2B1%7D+%2B+C_%7B%5Ceta%2Cs%7D+N%5E%7Bs%2B1%7D+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2814%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty n^s \eta(n/N) = -\frac{B_{s+1}}{s+1} + C_{\eta,s} N^{s+1} + O(\frac{1}{N}) \ \ \ \ \ (14)" class="latex" /></p> <p><a name="zeta-asym-s"></a> for any fixed <img src="https://s0.wp.com/latex.php?latex=%7Bs%3D1%2C2%2C3%2C%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%3D1%2C2%2C3%2C%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%3D1%2C2%2C3%2C%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s=1,2,3,\ldots}" class="latex" /> where <img src="https://s0.wp.com/latex.php?latex=%7BC_%7B%5Ceta%2Cs%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BC_%7B%5Ceta%2Cs%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BC_%7B%5Ceta%2Cs%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{C_{\eta,s}}" class="latex" /> is the Archimedean factor <a name="ceta"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+C_%7B%5Ceta%2Cs%7D+%3A%3D+%5Cint_0%5E%5Cinfty+x%5Es+%5Ceta%28x%29%5C+dx+%5C+%5C+%5C+%5C+%5C+%2815%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+C_%7B%5Ceta%2Cs%7D+%3A%3D+%5Cint_0%5E%5Cinfty+x%5Es+%5Ceta%28x%29%5C+dx+%5C+%5C+%5C+%5C+%5C+%2815%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+C_%7B%5Ceta%2Cs%7D+%3A%3D+%5Cint_0%5E%5Cinfty+x%5Es+%5Ceta%28x%29%5C+dx+%5C+%5C+%5C+%5C+%5C+%2815%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle C_{\eta,s} := \int_0^\infty x^s \eta(x)\ dx \ \ \ \ \ (15)" class="latex" /></p> <p><a name="ceta"></a> (which is also essentially the <a href="http://en.wikipedia.org/wiki/Mellin_transform">Mellin transform</a> of <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta}" class="latex" />). Thus we see that the values <a href="#zeta-1">(4)</a>, <a href="#zeta-2">(5)</a>, <a href="#zeta-3">(6)</a>, <a href="#zeta-s">(7)</a> obtained by analytic continuation are nothing more than the constant terms of the asymptotic expansion of the <em>smoothed</em> partial sums. This is not a coincidence; we will explain the equivalence of these two interpretations of such sums (in the model case when the analytic continuation has only finitely many poles and does not grow too fast at infinity) below the fold.</p> <p>This interpretation clears up the apparent inconsistencies alluded to earlier. For instance, the sum <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n+%3D+1+%2B+2+%2B+3+%2B+%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n+%3D+1+%2B+2+%2B+3+%2B+%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n+%3D+1+%2B+2+%2B+3+%2B+%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{n=1}^\infty n = 1 + 2 + 3 + \ldots}" class="latex" /> consists only of non-negative terms, as does its smoothed partial sums <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n+%5Ceta%28n%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n+%5Ceta%28n%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n+%5Ceta%28n%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{n=1}^\infty n \eta(n/N)}" class="latex" /> (if <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta}" class="latex" /> is non-negative). Comparing this with <a href="#zeta-asym-2">(12)</a>, we see that this forces the highest-order term <img src="https://s0.wp.com/latex.php?latex=%7BC_%7B%5Ceta%2C1%7D+N%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BC_%7B%5Ceta%2C1%7D+N%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BC_%7B%5Ceta%2C1%7D+N%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{C_{\eta,1} N^2}" class="latex" /> to be non-negative (as indeed it is), but does not prohibit the <em>lower-order</em> constant term <img src="https://s0.wp.com/latex.php?latex=%7B-%5Cfrac%7B1%7D%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B-%5Cfrac%7B1%7D%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B-%5Cfrac%7B1%7D%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{-\frac{1}{12}}" class="latex" /> from being negative (which of course it is).</p> <p>Similarly, if we add together <a href="#zeta-asym-2">(12)</a> and <a href="#zeta-asym-1">(11)</a> we obtain <a name="zeta-asym-a"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28n%2B1%29+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7B7%7D%7B12%7D+%2B+C_%7B%5Ceta%2C1%7D+N%5E2+%2B+C_%7B%5Ceta%2C0%7D+N+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2816%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28n%2B1%29+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7B7%7D%7B12%7D+%2B+C_%7B%5Ceta%2C1%7D+N%5E2+%2B+C_%7B%5Ceta%2C0%7D+N+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2816%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28n%2B1%29+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7B7%7D%7B12%7D+%2B+C_%7B%5Ceta%2C1%7D+N%5E2+%2B+C_%7B%5Ceta%2C0%7D+N+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29+%5C+%5C+%5C+%5C+%5C+%2816%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty (n+1) \eta(n/N) = -\frac{7}{12} + C_{\eta,1} N^2 + C_{\eta,0} N + O(\frac{1}{N}) \ \ \ \ \ (16)" class="latex" /></p> <p><a name="zeta-asym-a"></a> while if we subtract <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" /> from <a href="#zeta-asym-2">(12)</a> we obtain <a name="zeta-asym-b"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D2%7D%5E%5Cinfty+n+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7B13%7D%7B12%7D+%2B+C_%7B%5Ceta%2C1%7D+N%5E2+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29.+%5C+%5C+%5C+%5C+%5C+%2817%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D2%7D%5E%5Cinfty+n+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7B13%7D%7B12%7D+%2B+C_%7B%5Ceta%2C1%7D+N%5E2+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29.+%5C+%5C+%5C+%5C+%5C+%2817%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D2%7D%5E%5Cinfty+n+%5Ceta%28n%2FN%29+%3D+-%5Cfrac%7B13%7D%7B12%7D+%2B+C_%7B%5Ceta%2C1%7D+N%5E2+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29.+%5C+%5C+%5C+%5C+%5C+%2817%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=2}^\infty n \eta(n/N) = -\frac{13}{12} + C_{\eta,1} N^2 + O(\frac{1}{N}). \ \ \ \ \ (17)" class="latex" /></p> <p><a name="zeta-asym-b"></a> These two asymptotics are not inconsistent with each other; indeed, if we shift the index of summation in <a href="#zeta-asym-b">(17)</a>, we can write <a name="zetap"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D2%7D%5E%5Cinfty+n+%5Ceta%28n%2FN%29+%3D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28n%2B1%29+%5Ceta%28%28n%2B1%29%2FN%29+%5C+%5C+%5C+%5C+%5C+%2818%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D2%7D%5E%5Cinfty+n+%5Ceta%28n%2FN%29+%3D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28n%2B1%29+%5Ceta%28%28n%2B1%29%2FN%29+%5C+%5C+%5C+%5C+%5C+%2818%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D2%7D%5E%5Cinfty+n+%5Ceta%28n%2FN%29+%3D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%28n%2B1%29+%5Ceta%28%28n%2B1%29%2FN%29+%5C+%5C+%5C+%5C+%5C+%2818%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=2}^\infty n \eta(n/N) = \sum_{n=1}^\infty (n+1) \eta((n+1)/N) \ \ \ \ \ (18)" class="latex" /></p> <p><a name="zetap"></a> and so we now see that the discrepancy between the two sums in <a href="#sam1">(8)</a>, <a href="#sam2">(9)</a> come from the shifting of the cutoff <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%28n%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%28n%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%28n%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta(n/N)}" class="latex" />, which is invisible in the formal expressions in <a href="#sam1">(8)</a>, <a href="#sam2">(9)</a> but become manifestly present in the smoothed sum formulation.</p> <blockquote><p><b>Exercise 3</b> By Taylor expanding <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%28n%2B1%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%28n%2B1%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%28n%2B1%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta(n+1/N)}" class="latex" /> and using <a href="#zeta-asym-1">(11)</a>, <a href="#zetap">(18)</a> show that <a href="#zeta-asym-a">(16)</a> and <a href="#zeta-asym-b">(17)</a> are indeed consistent with each other, and in particular one can deduce the latter from the former.</p></blockquote> <p><span id="more-3643"></span></p> <p align="center"><b> — 1. Smoothed asymptotics — </b></p> <p>We now prove <a href="#zeta-asym-1">(11)</a>, <a href="#zeta-asym-2">(12)</a>, <a href="#zeta-asym-3">(13)</a>, <a href="#zeta-asym-s">(14)</a>. We will prove the first few asymptotics by <em>ad hoc</em> methods, but then switch to the systematic method of the <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula">Euler-Maclaurin formula</a> to establish the general case.</p> <p>For sake of argument we shall assume that the smooth cutoff <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%3A+%7B%5Cbf+R%7D%5E%2B+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%3A+%7B%5Cbf+R%7D%5E%2B+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%3A+%7B%5Cbf+R%7D%5E%2B+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta: {\bf R}^+ \rightarrow {\bf R}}" class="latex" /> is supported in the interval <img src="https://s0.wp.com/latex.php?latex=%7B%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{[0,1]}" class="latex" /> (the general case is similar, and can also be deduced from this case by redefining the <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" /> parameter). Thus the sum <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+x%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+x%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+x%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{n=1}^\infty \eta(n/N) x^s}" class="latex" /> is now only non-trivial in the range <img src="https://s0.wp.com/latex.php?latex=%7Bn+%5Cleq+N%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn+%5Cleq+N%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn+%5Cleq+N%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n \leq N}" class="latex" />.</p> <p>To establish <a href="#zeta-asym-1">(11)</a>, we shall exploit the <a href="http://en.wikipedia.org/wiki/Trapezoidal_rule">trapezoidal rule</a>. For any smooth function <img src="https://s0.wp.com/latex.php?latex=%7Bf%3A+%7B%5Cbf+R%7D+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%3A+%7B%5Cbf+R%7D+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%3A+%7B%5Cbf+R%7D+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f: {\bf R} \rightarrow {\bf R}}" class="latex" />, and on an interval <img src="https://s0.wp.com/latex.php?latex=%7B%5Bn%2Cn%2B1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Bn%2Cn%2B1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Bn%2Cn%2B1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{[n,n+1]}" class="latex" />, we see from Taylor expansion that</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B%5Ctheta%29+%3D+f%28n%29+%2B+%5Ctheta+f%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B%5Ctheta%29+%3D+f%28n%29+%2B+%5Ctheta+f%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B%5Ctheta%29+%3D+f%28n%29+%2B+%5Ctheta+f%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle f(n+\theta) = f(n) + \theta f'(n) + O( \|f\|_{\dot C^2} )" class="latex" /></p> <p>for any <img src="https://s0.wp.com/latex.php?latex=%7B0+%5Cleq+%5Ctheta+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B0+%5Cleq+%5Ctheta+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B0+%5Cleq+%5Ctheta+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{0 \leq \theta \leq 1}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%3A%3D+%5Csup_%7Bx+%5Cin+%7B%5Cbf+R%7D%7D+%7Cf%27%27%28x%29%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%3A%3D+%5Csup_%7Bx+%5Cin+%7B%5Cbf+R%7D%7D+%7Cf%27%27%28x%29%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%3A%3D+%5Csup_%7Bx+%5Cin+%7B%5Cbf+R%7D%7D+%7Cf%27%27%28x%29%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\|f\|_{\dot C^2} := \sup_{x \in {\bf R}} |f''(x)|}" class="latex" />. In particular we have</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B1%29+%3D+f%28n%29+%2B+f%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B1%29+%3D+f%28n%29+%2B+f%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B1%29+%3D+f%28n%29+%2B+f%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle f(n+1) = f(n) + f'(n) + O( \|f\|_{\dot C^2} )" class="latex" /></p> <p>and</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29%3B&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29%3B&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29%3B&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_n^{n+1} f(x)\ dx = f(n) + \frac{1}{2} f'(n) + O( \|f\|_{\dot C^2} );" class="latex" /></p> <p>eliminating <img src="https://s0.wp.com/latex.php?latex=%7Bf%27%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%27%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%27%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f'(n)}" class="latex" />, we conclude that</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%28n%2B1%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%28n%2B1%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%28n%2B1%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_n^{n+1} f(x)\ dx = \frac{1}{2} f(n) + \frac{1}{2} f(n+1) + O( \|f\|_{\dot C^2} )." class="latex" /></p> <p>Summing in <img src="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n}" class="latex" />, we conclude the trapezoidal rule</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%280%29+%2B+f%281%29+%2B+%5Cldots+%2B+f%28N-1%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%28N%29+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%280%29+%2B+f%281%29+%2B+%5Cldots+%2B+f%28N-1%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%28N%29+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%280%29+%2B+f%281%29+%2B+%5Cldots+%2B+f%28N-1%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%28N%29+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^N f(x)\ dx = \frac{1}{2} f(0) + f(1) + \ldots + f(N-1) + \frac{1}{2} f(N) " class="latex" /></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+N+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+N+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+N+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E2%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle + O( N \|f\|_{\dot C^2} )." class="latex" /></p> <p>We apply this with <img src="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f(x) := \eta(x/N)}" class="latex" />, which has a <img src="https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\dot C^2}" class="latex" /> norm of <img src="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(1/N^2)}" class="latex" /> from the chain rule, and conclude that</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+%5Ceta%28x%2FN%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+%2B+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+%2B+O%28+1%2FN+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+%5Ceta%28x%2FN%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+%2B+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+%2B+O%28+1%2FN+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+%5Ceta%28x%2FN%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+%2B+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ceta%28n%2FN%29+%2B+O%28+1%2FN+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^N \eta(x/N)\ dx = \frac{1}{2} + \sum_{n=1}^\infty \eta(n/N) + O( 1/N )." class="latex" /></p> <p>But from <a href="#ceta">(15)</a> and a change of variables, the left-hand side is just <img src="https://s0.wp.com/latex.php?latex=%7BN+C_%7B%5Ceta%2C0%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN+C_%7B%5Ceta%2C0%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN+C_%7B%5Ceta%2C0%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N C_{\eta,0}}" class="latex" />. This gives <a href="#zeta-asym-1">(11)</a>.</p> <p>The same argument does not quite work with <a href="#zeta-asym-2">(12)</a>; one would like to now set <img src="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f(x) := x \eta(x/N)}" class="latex" />, but the <img src="https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\dot C^2}" class="latex" /> norm is now too large (<img src="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(1/N)}" class="latex" /> instead of <img src="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(1/N^2)}" class="latex" />). To get around this we have to refine the trapezoidal rule by performing the more precise Taylor expansion</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B%5Ctheta%29+%3D+f%28n%29+%2B+%5Ctheta+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+%5Ctheta%5E2+f%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B%5Ctheta%29+%3D+f%28n%29+%2B+%5Ctheta+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+%5Ctheta%5E2+f%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B%5Ctheta%29+%3D+f%28n%29+%2B+%5Ctheta+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+%5Ctheta%5E2+f%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle f(n+\theta) = f(n) + \theta f'(n) + \frac{1}{2} \theta^2 f''(n) + O( \|f\|_{\dot C^3} )" class="latex" /></p> <p>where <img src="https://s0.wp.com/latex.php?latex=%7B%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%3A%3D+%5Csup_%7Bx+%5Cin+%7B%5Cbf+R%7D%7D+%7Cf%27%27%27%28x%29%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%3A%3D+%5Csup_%7Bx+%5Cin+%7B%5Cbf+R%7D%7D+%7Cf%27%27%27%28x%29%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%3A%3D+%5Csup_%7Bx+%5Cin+%7B%5Cbf+R%7D%7D+%7Cf%27%27%27%28x%29%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\|f\|_{\dot C^3} := \sup_{x \in {\bf R}} |f'''(x)|}" class="latex" />. Now we have</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B1%29+%3D+f%28n%29+%2B+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B1%29+%3D+f%28n%29+%2B+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B1%29+%3D+f%28n%29+%2B+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle f(n+1) = f(n) + f'(n) + \frac{1}{2} f''(n) + O( \|f\|_{\dot C^3} )" class="latex" /></p> <p>and</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B6%7D+f%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B6%7D+f%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B6%7D+f%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_n^{n+1} f(x)\ dx = f(n) + \frac{1}{2} f'(n) + \frac{1}{6} f''(n) + O( \|f\|_{\dot C^3} )." class="latex" /></p> <p>We cannot simultaneously eliminate both <img src="https://s0.wp.com/latex.php?latex=%7Bf%27%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%27%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%27%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f'(n)}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bf%27%27%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%27%27%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%27%27%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f''(n)}" class="latex" />. However, using the additional Taylor expansion</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%27%28n%2B1%29+%3D+f%27%28n%29+%2B+f%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%27%28n%2B1%29+%3D+f%27%28n%29+%2B+f%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%27%28n%2B1%29+%3D+f%27%28n%29+%2B+f%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle f'(n+1) = f'(n) + f''(n) + O( \|f\|_{\dot C^3} )" class="latex" /></p> <p>one obtains</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%28n%2B1%29+%2B+%5Cfrac%7B1%7D%7B12%7D+%28f%27%28n%29+-+f%27%28n%2B1%29%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%28n%2B1%29+%2B+%5Cfrac%7B1%7D%7B12%7D+%28f%27%28n%29+-+f%27%28n%2B1%29%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%28n%2B1%29+%2B+%5Cfrac%7B1%7D%7B12%7D+%28f%27%28n%29+-+f%27%28n%2B1%29%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_n^{n+1} f(x)\ dx = \frac{1}{2} f(n) + \frac{1}{2} f(n+1) + \frac{1}{12} (f'(n) - f'(n+1))" class="latex" /></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle + O( \|f\|_{\dot C^3} )" class="latex" /></p> <p>and thus on summing in <img src="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n}" class="latex" />, and assuming that <img src="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f}" class="latex" /> vanishes to second order at <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" />, one has (by <a href="http://en.wikipedia.org/wiki/Telescoping_series">telescoping series</a>)</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%280%29+%2B+%5Cfrac%7B1%7D%7B12%7D+f%27%280%29+%2B+%5Csum_%7Bn%3D1%7D%5EN+f%28n%29+%2B+O%28+N+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%280%29+%2B+%5Cfrac%7B1%7D%7B12%7D+f%27%280%29+%2B+%5Csum_%7Bn%3D1%7D%5EN+f%28n%29+%2B+O%28+N+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%280%29+%2B+%5Cfrac%7B1%7D%7B12%7D+f%27%280%29+%2B+%5Csum_%7Bn%3D1%7D%5EN+f%28n%29+%2B+O%28+N+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^N f(x)\ dx = \frac{1}{2} f(0) + \frac{1}{12} f'(0) + \sum_{n=1}^N f(n) + O( N \|f\|_{\dot C^3} )." class="latex" /></p> <p>We apply this with <img src="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f(x) := x \eta(x/N)}" class="latex" />. After a few applications of the chain rule and product rule, we see that <img src="https://s0.wp.com/latex.php?latex=%7B%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%3D+O%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%3D+O%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5C%7Cf%5C%7C_%7B%5Cdot+C%5E3%7D+%3D+O%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\|f\|_{\dot C^3} = O(1/N^2)}" class="latex" />; also, <img src="https://s0.wp.com/latex.php?latex=%7Bf%280%29%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%280%29%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%280%29%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f(0)=0}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7Bf%27%280%29%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%27%280%29%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%27%280%29%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f'(0)=1}" class="latex" />, and <img src="https://s0.wp.com/latex.php?latex=%7B%5Cint_0%5EN+f%28x%29%5C+dx+%3D+N%5E2+C_%7B%5Ceta%2C1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cint_0%5EN+f%28x%29%5C+dx+%3D+N%5E2+C_%7B%5Ceta%2C1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cint_0%5EN+f%28x%29%5C+dx+%3D+N%5E2+C_%7B%5Ceta%2C1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\int_0^N f(x)\ dx = N^2 C_{\eta,1}}" class="latex" />. This gives <a href="#zeta-asym-2">(12)</a>.</p> <p>The proof of <a href="#zeta-asym-3">(13)</a> is similar. With a fourth order Taylor expansion, the above arguments give</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B1%29+%3D+f%28n%29+%2B+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%27%28n%29+%2B+%5Cfrac%7B1%7D%7B6%7D+f%27%27%27%28x%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29%2C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B1%29+%3D+f%28n%29+%2B+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%27%28n%29+%2B+%5Cfrac%7B1%7D%7B6%7D+f%27%27%27%28x%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29%2C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28n%2B1%29+%3D+f%28n%29+%2B+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%27%28n%29+%2B+%5Cfrac%7B1%7D%7B6%7D+f%27%27%27%28x%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29%2C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle f(n+1) = f(n) + f'(n) + \frac{1}{2} f''(n) + \frac{1}{6} f'''(x) + O( \|f\|_{\dot C^4} )," class="latex" /></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B6%7D+f%27%27%28n%29+%2B+%5Cfrac%7B1%7D%7B24%7D+f%27%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B6%7D+f%27%27%28n%29+%2B+%5Cfrac%7B1%7D%7B24%7D+f%27%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%28n%29+%2B+%5Cfrac%7B1%7D%7B6%7D+f%27%27%28n%29+%2B+%5Cfrac%7B1%7D%7B24%7D+f%27%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_n^{n+1} f(x)\ dx = f(n) + \frac{1}{2} f'(n) + \frac{1}{6} f''(n) + \frac{1}{24} f'''(n) + O( \|f\|_{\dot C^4} )" class="latex" /></p> <p>and</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%27%28n%2B1%29+%3D+f%27%28n%29+%2B+f%27%27%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%27%28n%2B1%29+%3D+f%27%28n%29+%2B+f%27%27%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%27%28n%2B1%29+%3D+f%27%28n%29+%2B+f%27%27%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%27%27%27%28n%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle f'(n+1) = f'(n) + f''(n) + \frac{1}{2} f'''(n) + O( \|f\|_{\dot C^4} )." class="latex" /></p> <p>Here we have a minor miracle (equivalent to the vanishing of the third Bernoulli number <img src="https://s0.wp.com/latex.php?latex=%7BB_3%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB_3%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB_3%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B_3}" class="latex" />) that the <img src="https://s0.wp.com/latex.php?latex=%7Bf%27%27%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%27%27%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%27%27%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f'''}" class="latex" /> term is automatically eliminated when we eliminate the <img src="https://s0.wp.com/latex.php?latex=%7Bf%27%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%27%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%27%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f''}" class="latex" /> term, yielding</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%28n%2B1%29+%2B+%5Cfrac%7B1%7D%7B12%7D+%28f%27%28n%29+-+f%27%28n%2B1%29%29+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%28n%2B1%29+%2B+%5Cfrac%7B1%7D%7B12%7D+%28f%27%28n%29+-+f%27%28n%2B1%29%29+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%28n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+f%28n%2B1%29+%2B+%5Cfrac%7B1%7D%7B12%7D+%28f%27%28n%29+-+f%27%28n%2B1%29%29+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_n^{n+1} f(x)\ dx = \frac{1}{2} f(n) + \frac{1}{2} f(n+1) + \frac{1}{12} (f'(n) - f'(n+1)) " class="latex" /></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle + O( \|f\|_{\dot C^4} )" class="latex" /></p> <p>and thus</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%280%29+%2B+%5Cfrac%7B1%7D%7B12%7D+f%27%280%29+%2B+%5Csum_%7Bn%3D1%7D%5EN+f%28n%29+%2B+O%28+N+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%280%29+%2B+%5Cfrac%7B1%7D%7B12%7D+f%27%280%29+%2B+%5Csum_%7Bn%3D1%7D%5EN+f%28n%29+%2B+O%28+N+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%280%29+%2B+%5Cfrac%7B1%7D%7B12%7D+f%27%280%29+%2B+%5Csum_%7Bn%3D1%7D%5EN+f%28n%29+%2B+O%28+N+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E4%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^N f(x)\ dx = \frac{1}{2} f(0) + \frac{1}{12} f'(0) + \sum_{n=1}^N f(n) + O( N \|f\|_{\dot C^4} )." class="latex" /></p> <p>With <img src="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x%5E2+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x%5E2+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x%5E2+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f(x) := x^2 \eta(x/N)}" class="latex" />, the left-hand side is <img src="https://s0.wp.com/latex.php?latex=%7BN%5E3+C_%7B%5Ceta%2C2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%5E3+C_%7B%5Ceta%2C2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%5E3+C_%7B%5Ceta%2C2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N^3 C_{\eta,2}}" class="latex" />, the first two terms on the right-hand side vanish, and the <img src="https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E4%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E4%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E4%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\dot C^4}" class="latex" /> norm is <img src="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(1/N^2)}" class="latex" />, giving <a href="#zeta-asym-3">(13)</a>.</p> <p>Now we do the general case <a href="#zeta-asym-s">(14)</a>. We define the <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli numbers</a> <img src="https://s0.wp.com/latex.php?latex=%7BB_0%2C+B_1%2C+%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB_0%2C+B_1%2C+%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB_0%2C+B_1%2C+%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B_0, B_1, \ldots}" class="latex" /> recursively by the formula <a name="bak"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bk%3D0%7D%5E%7Bs-1%7D+%5Cbinom%7Bs%7D%7Bk%7D+B_k+%3D+s+%5C+%5C+%5C+%5C+%5C+%2819%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bk%3D0%7D%5E%7Bs-1%7D+%5Cbinom%7Bs%7D%7Bk%7D+B_k+%3D+s+%5C+%5C+%5C+%5C+%5C+%2819%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bk%3D0%7D%5E%7Bs-1%7D+%5Cbinom%7Bs%7D%7Bk%7D+B_k+%3D+s+%5C+%5C+%5C+%5C+%5C+%2819%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{k=0}^{s-1} \binom{s}{k} B_k = s \ \ \ \ \ (19)" class="latex" /></p> <p><a name="bak"></a> for all <img src="https://s0.wp.com/latex.php?latex=%7Bs+%3D1%2C2%2C%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs+%3D1%2C2%2C%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs+%3D1%2C2%2C%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s =1,2,\ldots}" class="latex" />, or equivalently</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+B_%7Bs-1%7D+%3A%3D+1+-+%5Cfrac%7Bs-1%7D%7B2%7D+B_%7Bs-2%7D+-+%5Cfrac%7B%28s-1%29%28s-2%29%7D%7B3%21%7D+B_%7Bs-3%7D+-+%5Cldots+-+%5Cfrac%7B1%7D%7Bs%7D+B_0.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+B_%7Bs-1%7D+%3A%3D+1+-+%5Cfrac%7Bs-1%7D%7B2%7D+B_%7Bs-2%7D+-+%5Cfrac%7B%28s-1%29%28s-2%29%7D%7B3%21%7D+B_%7Bs-3%7D+-+%5Cldots+-+%5Cfrac%7B1%7D%7Bs%7D+B_0.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+B_%7Bs-1%7D+%3A%3D+1+-+%5Cfrac%7Bs-1%7D%7B2%7D+B_%7Bs-2%7D+-+%5Cfrac%7B%28s-1%29%28s-2%29%7D%7B3%21%7D+B_%7Bs-3%7D+-+%5Cldots+-+%5Cfrac%7B1%7D%7Bs%7D+B_0.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle B_{s-1} := 1 - \frac{s-1}{2} B_{s-2} - \frac{(s-1)(s-2)}{3!} B_{s-3} - \ldots - \frac{1}{s} B_0." class="latex" /></p> <p>The first few values of <img src="https://s0.wp.com/latex.php?latex=%7BB_s%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB_s%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB_s%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B_s}" class="latex" /> can then be computed:</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+B_0%3D1%3B+B_1%3D1%2F2%3B+B_2%3D1%2F6%3B+B_3%3D0%3B+B_4%3D-1%2F30%3B+%5Cldots.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+B_0%3D1%3B+B_1%3D1%2F2%3B+B_2%3D1%2F6%3B+B_3%3D0%3B+B_4%3D-1%2F30%3B+%5Cldots.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+B_0%3D1%3B+B_1%3D1%2F2%3B+B_2%3D1%2F6%3B+B_3%3D0%3B+B_4%3D-1%2F30%3B+%5Cldots.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle B_0=1; B_1=1/2; B_2=1/6; B_3=0; B_4=-1/30; \ldots." class="latex" /></p> <p>From <a href="#bak">(19)</a> we see that <a name="bkak"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5B+P%5E%7B%28k%29%7D%281%29+-+P%5E%7B%28k%29%7D%280%29+%5D+%3D+P%27%281%29+%5C+%5C+%5C+%5C+%5C+%2820%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5B+P%5E%7B%28k%29%7D%281%29+-+P%5E%7B%28k%29%7D%280%29+%5D+%3D+P%27%281%29+%5C+%5C+%5C+%5C+%5C+%2820%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5B+P%5E%7B%28k%29%7D%281%29+-+P%5E%7B%28k%29%7D%280%29+%5D+%3D+P%27%281%29+%5C+%5C+%5C+%5C+%5C+%2820%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{k=0}^\infty \frac{B_k}{k!} [ P^{(k)}(1) - P^{(k)}(0) ] = P'(1) \ \ \ \ \ (20)" class="latex" /></p> <p><a name="bkak"></a> for any polynomial <img src="https://s0.wp.com/latex.php?latex=%7BP%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BP%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BP%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{P}" class="latex" /> (with <img src="https://s0.wp.com/latex.php?latex=%7BP%5E%7B%28k%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BP%5E%7B%28k%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BP%5E%7B%28k%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{P^{(k)}}" class="latex" /> being the <img src="https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k}" class="latex" />-fold derivative of <img src="https://s0.wp.com/latex.php?latex=%7BP%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BP%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BP%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{P}" class="latex" />); indeed, <a href="#bak">(19)</a> is precisely this identity with <img src="https://s0.wp.com/latex.php?latex=%7BP%28x%29+%3A%3D+x%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BP%28x%29+%3A%3D+x%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BP%28x%29+%3A%3D+x%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{P(x) := x^s}" class="latex" />, and the general case then follows by linearity.</p> <p>As <a href="#bkak">(20)</a> holds for all polynomials, it also holds for all formal power series (if we ignore convergence issues). If we then replace <img src="https://s0.wp.com/latex.php?latex=%7BP%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BP%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BP%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{P}" class="latex" /> by the formal power series</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+P%28x%29+%3D+e%5E%7Btx%7D+%3D+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+t%5Ek+%5Cfrac%7Bx%5Ek%7D%7Bk%21%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+P%28x%29+%3D+e%5E%7Btx%7D+%3D+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+t%5Ek+%5Cfrac%7Bx%5Ek%7D%7Bk%21%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+P%28x%29+%3D+e%5E%7Btx%7D+%3D+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+t%5Ek+%5Cfrac%7Bx%5Ek%7D%7Bk%21%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle P(x) = e^{tx} = \sum_{k=0}^\infty t^k \frac{x^k}{k!}" class="latex" /></p> <p>we conclude the formal power series (in <img src="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t}" class="latex" />) identity</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+t%5Ek+%28e%5Et-1%29+%3D+t+e%5Et&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+t%5Ek+%28e%5Et-1%29+%3D+t+e%5Et&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+t%5Ek+%28e%5Et-1%29+%3D+t+e%5Et&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{k=0}^\infty \frac{B_k}{k!} t^k (e^t-1) = t e^t" class="latex" /></p> <p>leading to the familiar generating function</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+t%5Ek+%3D+%5Cfrac%7Bt+e%5Et%7D%7Be%5Et-1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+t%5Ek+%3D+%5Cfrac%7Bt+e%5Et%7D%7Be%5Et-1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+t%5Ek+%3D+%5Cfrac%7Bt+e%5Et%7D%7Be%5Et-1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{k=0}^\infty \frac{B_k}{k!} t^k = \frac{t e^t}{e^t-1}" class="latex" /></p> <p>for the Bernoulli numbers.</p> <p>If we apply <a href="#bkak">(20)</a> with <img src="https://s0.wp.com/latex.php?latex=%7BP%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BP%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BP%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{P}" class="latex" /> equal to the antiderivative of another polynomial <img src="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Q}" class="latex" />, we conclude that</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5E1+Q%28x%29%5C+dx+%2B+%5Cfrac%7B1%7D%7B2%7D+%28Q%281%29-Q%280%29%29+%2B+%5Csum_%7Bk%3D2%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5BQ%5E%7B%28k-1%29%7D%281%29+-+Q%5E%7B%28k-1%29%7D%280%29%5D+%3D+Q%281%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5E1+Q%28x%29%5C+dx+%2B+%5Cfrac%7B1%7D%7B2%7D+%28Q%281%29-Q%280%29%29+%2B+%5Csum_%7Bk%3D2%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5BQ%5E%7B%28k-1%29%7D%281%29+-+Q%5E%7B%28k-1%29%7D%280%29%5D+%3D+Q%281%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5E1+Q%28x%29%5C+dx+%2B+%5Cfrac%7B1%7D%7B2%7D+%28Q%281%29-Q%280%29%29+%2B+%5Csum_%7Bk%3D2%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5BQ%5E%7B%28k-1%29%7D%281%29+-+Q%5E%7B%28k-1%29%7D%280%29%5D+%3D+Q%281%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^1 Q(x)\ dx + \frac{1}{2} (Q(1)-Q(0)) + \sum_{k=2}^\infty \frac{B_k}{k!} [Q^{(k-1)}(1) - Q^{(k-1)}(0)] = Q(1)" class="latex" /></p> <p>which we rearrange as the identity</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5E1+Q%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D%28Q%280%29%2BQ%281%29%29+-+%5Csum_%7Bk%3D2%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5BQ%5E%7B%28k-1%29%7D%281%29+-+Q%5E%7B%28k-1%29%7D%280%29%5D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5E1+Q%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D%28Q%280%29%2BQ%281%29%29+-+%5Csum_%7Bk%3D2%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5BQ%5E%7B%28k-1%29%7D%281%29+-+Q%5E%7B%28k-1%29%7D%280%29%5D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5E1+Q%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D%28Q%280%29%2BQ%281%29%29+-+%5Csum_%7Bk%3D2%7D%5E%5Cinfty+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5BQ%5E%7B%28k-1%29%7D%281%29+-+Q%5E%7B%28k-1%29%7D%280%29%5D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^1 Q(x)\ dx = \frac{1}{2}(Q(0)+Q(1)) - \sum_{k=2}^\infty \frac{B_k}{k!} [Q^{(k-1)}(1) - Q^{(k-1)}(0)]" class="latex" /></p> <p>which can be viewed as a precise version of the trapezoidal rule in the polynomial case. Note that if <img src="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Q}" class="latex" /> has degree <img src="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d}" class="latex" />, the only the summands with <img src="https://s0.wp.com/latex.php?latex=%7B2+%5Cleq+k+%5Cleq+d%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2+%5Cleq+k+%5Cleq+d%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2+%5Cleq+k+%5Cleq+d%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2 \leq k \leq d}" class="latex" /> can be non-vanishing.</p> <p>Now let <img src="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f}" class="latex" /> be a smooth function. We have a Taylor expansion</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28x%29+%3D+Q%28x%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28x%29+%3D+Q%28x%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28x%29+%3D+Q%28x%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle f(x) = Q(x) + O( \|f\|_{\dot C^{s+2}} )" class="latex" /></p> <p>for <img src="https://s0.wp.com/latex.php?latex=%7B0+%5Cleq+x+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B0+%5Cleq+x+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B0+%5Cleq+x+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{0 \leq x \leq 1}" class="latex" /> and some polynomial <img src="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Q}" class="latex" /> of degree at most <img src="https://s0.wp.com/latex.php?latex=%7Bs%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s+1}" class="latex" />; also</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%5E%7B%28k-1%29%7D%28x%29+%3D+Q%5E%7B%28k-1%29%7D%28x%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%5E%7B%28k-1%29%7D%28x%29+%3D+Q%5E%7B%28k-1%29%7D%28x%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%5E%7B%28k-1%29%7D%28x%29+%3D+Q%5E%7B%28k-1%29%7D%28x%29+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle f^{(k-1)}(x) = Q^{(k-1)}(x) + O( \|f\|_{\dot C^{s+2}} )" class="latex" /></p> <p>for <img src="https://s0.wp.com/latex.php?latex=%7B0+%5Cleq+x+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B0+%5Cleq+x+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B0+%5Cleq+x+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{0 \leq x \leq 1}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bk+%5Cleq+s%2B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk+%5Cleq+s%2B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk+%5Cleq+s%2B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k \leq s+2}" class="latex" />. We conclude that</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5E1+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D%28f%280%29%2Bf%281%29%29+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5E1+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D%28f%280%29%2Bf%281%29%29+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5E1+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D%28f%280%29%2Bf%281%29%29+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^1 f(x)\ dx = \frac{1}{2}(f(0)+f(1)) " class="latex" /></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%5Csum_%7Bk%3D2%7D%5E%7Bs%2B1%7D+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5Bf%5E%7B%28k-1%29%7D%281%29+-+f%5E%7B%28k-1%29%7D%280%29%5D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%5Csum_%7Bk%3D2%7D%5E%7Bs%2B1%7D+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5Bf%5E%7B%28k-1%29%7D%281%29+-+f%5E%7B%28k-1%29%7D%280%29%5D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%5Csum_%7Bk%3D2%7D%5E%7Bs%2B1%7D+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5Bf%5E%7B%28k-1%29%7D%281%29+-+f%5E%7B%28k-1%29%7D%280%29%5D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle - \sum_{k=2}^{s+1} \frac{B_k}{k!} [f^{(k-1)}(1) - f^{(k-1)}(0)]" class="latex" /></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle + O( \|f\|_{\dot C^{s+2}} )." class="latex" /></p> <p>Translating this by an arbitrary integer <img src="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n}" class="latex" /> (which does not affect the <img src="https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E%7Bs%2B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E%7Bs%2B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E%7Bs%2B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\dot C^{s+2}}" class="latex" /> norm), we obtain <a name="qx"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D%28f%28n%29%2Bf%28n%2B1%29%29+%5C+%5C+%5C+%5C+%5C+%2821%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D%28f%28n%29%2Bf%28n%2B1%29%29+%5C+%5C+%5C+%5C+%5C+%2821%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_n%5E%7Bn%2B1%7D+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D%28f%28n%29%2Bf%28n%2B1%29%29+%5C+%5C+%5C+%5C+%5C+%2821%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_n^{n+1} f(x)\ dx = \frac{1}{2}(f(n)+f(n+1)) \ \ \ \ \ (21)" class="latex" /></p> <p><a name="qx"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%5Csum_%7Bk%3D2%7D%5E%7Bs%2B1%7D+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5Bf%5E%7B%28k-1%29%7D%28n%2B1%29+-+f%5E%7B%28k-1%29%7D%28n%29%5D+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29.+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%5Csum_%7Bk%3D2%7D%5E%7Bs%2B1%7D+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5Bf%5E%7B%28k-1%29%7D%28n%2B1%29+-+f%5E%7B%28k-1%29%7D%28n%29%5D+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29.+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%5Csum_%7Bk%3D2%7D%5E%7Bs%2B1%7D+%5Cfrac%7BB_k%7D%7Bk%21%7D+%5Bf%5E%7B%28k-1%29%7D%28n%2B1%29+-+f%5E%7B%28k-1%29%7D%28n%29%5D+%2B+O%28+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29.+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle - \sum_{k=2}^{s+1} \frac{B_k}{k!} [f^{(k-1)}(n+1) - f^{(k-1)}(n)] + O( \|f\|_{\dot C^{s+2}} ). " class="latex" /></p> <p>Summing the telescoping series, and assuming that <img src="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f}" class="latex" /> vanishes to a sufficiently high order at <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" />, we conclude the <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula">Euler-Maclaurin formula</a> <a name="salami"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%280%29+%2B+%5Csum_%7Bn%3D1%7D%5EN+f%28n%29+%2B+%5Csum_%7Bk%3D2%7D%5E%7Bs%2B1%7D+%5Cfrac%7BB_k%7D%7Bk%21%7D+f%5E%7B%28k-1%29%7D%280%29+%5C+%5C+%5C+%5C+%5C+%2822%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%280%29+%2B+%5Csum_%7Bn%3D1%7D%5EN+f%28n%29+%2B+%5Csum_%7Bk%3D2%7D%5E%7Bs%2B1%7D+%5Cfrac%7BB_k%7D%7Bk%21%7D+f%5E%7B%28k-1%29%7D%280%29+%5C+%5C+%5C+%5C+%5C+%2822%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5EN+f%28x%29%5C+dx+%3D+%5Cfrac%7B1%7D%7B2%7D+f%280%29+%2B+%5Csum_%7Bn%3D1%7D%5EN+f%28n%29+%2B+%5Csum_%7Bk%3D2%7D%5E%7Bs%2B1%7D+%5Cfrac%7BB_k%7D%7Bk%21%7D+f%5E%7B%28k-1%29%7D%280%29+%5C+%5C+%5C+%5C+%5C+%2822%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^N f(x)\ dx = \frac{1}{2} f(0) + \sum_{n=1}^N f(n) + \sum_{k=2}^{s+1} \frac{B_k}{k!} f^{(k-1)}(0) \ \ \ \ \ (22)" class="latex" /></p> <p><a name="salami"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+N+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29.+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+N+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29.+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+N+%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%29.+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle + O( N \|f\|_{\dot C^{s+2}} ). " class="latex" /></p> <p>We apply this with <img src="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x%5Es+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x%5Es+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x%5Es+%5Ceta%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f(x) := x^s \eta(x/N)}" class="latex" />. The left-hand side is <img src="https://s0.wp.com/latex.php?latex=%7BC_%7B%5Ceta%2Cs%7D+N%5E%7Bs%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BC_%7B%5Ceta%2Cs%7D+N%5E%7Bs%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BC_%7B%5Ceta%2Cs%7D+N%5E%7Bs%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{C_{\eta,s} N^{s+1}}" class="latex" />. All the terms in the sum vanish except for the <img src="https://s0.wp.com/latex.php?latex=%7Bk%3Ds%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk%3Ds%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk%3Ds%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k=s+1}" class="latex" /> term, which is <img src="https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7BB_%7Bs%2B1%7D%7D%7Bs%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7BB_%7Bs%2B1%7D%7D%7Bs%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7BB_%7Bs%2B1%7D%7D%7Bs%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\frac{B_{s+1}}{s+1}}" class="latex" />. Finally, from many applications of the product rule and chain rule (or by viewing <img src="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3D+N%5Es+g%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3D+N%5Es+g%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3D+N%5Es+g%28x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f(x) = N^s g(x/N)}" class="latex" /> where <img src="https://s0.wp.com/latex.php?latex=%7Bg%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bg%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bg%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{g}" class="latex" /> is the smooth function <img src="https://s0.wp.com/latex.php?latex=%7Bg%28x%29+%3A%3D+x%5Es+%5Ceta%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bg%28x%29+%3A%3D+x%5Es+%5Ceta%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bg%28x%29+%3A%3D+x%5Es+%5Ceta%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{g(x) := x^s \eta(x)}" class="latex" />) we see that <img src="https://s0.wp.com/latex.php?latex=%7B%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%3D+O%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%3D+O%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5C%7Cf%5C%7C_%7B%5Cdot+C%5E%7Bs%2B2%7D%7D+%3D+O%281%2FN%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\|f\|_{\dot C^{s+2}} = O(1/N^2)}" class="latex" />, and the claim <a href="#zeta-asym-s">(14)</a> follows.</p> <blockquote><p><b>Remark 4</b> <a name="nab"></a> By using a higher regularity norm than the <img src="https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E%7Bs%2B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E%7Bs%2B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5E%7Bs%2B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\dot C^{s+2}}" class="latex" /> norm, we see that the error term <img src="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(1/N)}" class="latex" /> can in fact be improved to <img src="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5EB%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5EB%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%281%2FN%5EB%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(1/N^B)}" class="latex" /> for any fixed <img src="https://s0.wp.com/latex.php?latex=%7BB%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B>0}" class="latex" />, if <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta}" class="latex" /> is sufficiently smooth.</p></blockquote> <blockquote><p><b>Exercise 5</b> Use <a href="#qx">(21)</a> to derive Faulhaber’s formula <a href="#faul">(10)</a>. Note how the presence of boundary terms at <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" /> cause the right-hand side of <a href="#faul">(10)</a> to be quite different from the right-hand side of <a href="#zeta-asym-s">(14)</a>; thus we see how non-smooth partial summation creates artefacts that can completely obscure the smoothed asymptotics.</p></blockquote> <p align="center"><b> — 2. Connection with analytic continuation — </b></p> <p>Now we connect the interpretation of divergent series as the constant term of smoothed partial sum asymptotics, with the more traditional interpretation via analytic continuation. For sake of concreteness we shall just discuss the situation with the Riemann zeta function series <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{n=1}^\infty \frac{1}{n^s}}" class="latex" />, though the connection extends to far more general series than just this one.</p> <p>In the previous section, we have computed asymptotics for the partial sums</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \frac{1}{n^s} \eta(n/N)" class="latex" /></p> <p>when <img src="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s}" class="latex" /> is a negative integer. A key point (which was somewhat glossed over in the above analysis) was that the function <img src="https://s0.wp.com/latex.php?latex=%7Bx%5E%7B-s%7D+%5Ceta%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%5E%7B-s%7D+%5Ceta%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%5E%7B-s%7D+%5Ceta%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x^{-s} \eta(x)}" class="latex" /> was smooth, even at the origin; this was implicitly used to bound various <img src="https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5Ek%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5Ek%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cdot+C%5Ek%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\dot C^k}" class="latex" /> norms in the error terms.</p> <p>Now suppose that <img src="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s}" class="latex" /> is a complex number with <img src="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29%3C1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29%3C1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29%3C1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\hbox{Re}(s)<1}" class="latex" />, which is not necessarily a negative integer. Then <img src="https://s0.wp.com/latex.php?latex=%7Bx%5E%7B-s%7D+%5Ceta%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%5E%7B-s%7D+%5Ceta%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%5E%7B-s%7D+%5Ceta%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x^{-s} \eta(x)}" class="latex" /> becomes singular at the origin, and the above asymptotic analysis is not directly applicable. However, if one instead considers the telescoped partial sum</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%282n%2FN%29%2C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%282n%2FN%29%2C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%282n%2FN%29%2C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \frac{1}{n^s} \eta(n/N) - \sum_{n=1}^\infty \frac{1}{n^s} \eta(2n/N)," class="latex" /></p> <p>with <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta}" class="latex" /> equal to <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" /> near the origin, then by applying <a href="#salami">(22)</a> to the function <img src="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x%5E%7B-s%7D+%5Ceta%28x%2FN%29+-+x%5E%7B-s%7D+%5Ceta%282x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x%5E%7B-s%7D+%5Ceta%28x%2FN%29+-+x%5E%7B-s%7D+%5Ceta%282x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%3A%3D+x%5E%7B-s%7D+%5Ceta%28x%2FN%29+-+x%5E%7B-s%7D+%5Ceta%282x%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f(x) := x^{-s} \eta(x/N) - x^{-s} \eta(2x/N)}" class="latex" /> (which vanishes near the origin, and is now smooth everywhere), we soon obtain the asymptotic <a name="haloo"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%282n%2FN%29+%5C+%5C+%5C+%5C+%5C+%2823%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%282n%2FN%29+%5C+%5C+%5C+%5C+%5C+%2823%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%282n%2FN%29+%5C+%5C+%5C+%5C+%5C+%2823%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \frac{1}{n^s} \eta(n/N) - \sum_{n=1}^\infty \frac{1}{n^s} \eta(2n/N) \ \ \ \ \ (23)" class="latex" /></p> <p><a name="haloo"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+C_%7B%5Ceta%2C-s%7D+%28N%5E%7B1-s%7D+-+%28N%2F2%29%5E%7B1-s%7D%29+%2B+O%281%2FN%29.+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+C_%7B%5Ceta%2C-s%7D+%28N%5E%7B1-s%7D+-+%28N%2F2%29%5E%7B1-s%7D%29+%2B+O%281%2FN%29.+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+C_%7B%5Ceta%2C-s%7D+%28N%5E%7B1-s%7D+-+%28N%2F2%29%5E%7B1-s%7D%29+%2B+O%281%2FN%29.+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle = C_{\eta,-s} (N^{1-s} - (N/2)^{1-s}) + O(1/N). " class="latex" /></p> <p>Applying this with <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" /> equal to a power of two and summing the telescoping series, one concludes that <a name="snag"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+%3D+%5Czeta%28s%29+%2B+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D+%2B+O%281%2FN%29+%5C+%5C+%5C+%5C+%5C+%2824%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+%3D+%5Czeta%28s%29+%2B+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D+%2B+O%281%2FN%29+%5C+%5C+%5C+%5C+%5C+%2824%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+%3D+%5Czeta%28s%29+%2B+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D+%2B+O%281%2FN%29+%5C+%5C+%5C+%5C+%5C+%2824%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \frac{1}{n^s} \eta(n/N) = \zeta(s) + C_{\eta,-s} N^{1-s} + O(1/N) \ \ \ \ \ (24)" class="latex" /></p> <p><a name="snag"></a> for some complex number <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(s)}" class="latex" /> which is basically the sum of the various <img src="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%281%2FN%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(1/N)}" class="latex" /> terms appearing in <a href="#haloo">(23)</a>. By modifying the above arguments, it is not difficult to extend this asymptotic to other numbers than powers of two, and to show that <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(s)}" class="latex" /> is independent of the choice of cutoff <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta}" class="latex" />.</p> <p>From <a href="#snag">(24)</a> we have</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29+%3D+%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D%2C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29+%3D+%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D%2C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29+%3D+%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D%2C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \zeta(s) = \lim_{N \rightarrow \infty} \sum_{n=1}^\infty \frac{1}{n^s} \eta(n/N) - C_{\eta,-s} N^{1-s}," class="latex" /></p> <p>which can be viewed as a definition of <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta}" class="latex" /> in the region <img src="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29%3C1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29%3C1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29%3C1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\hbox{Re}(s)<1}" class="latex" />. For instance, from <a href="#zeta-asym-s">(14)</a>, we have now proven <a href="#zeta-bern">(3)</a> with this definition of <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(s)}" class="latex" />. However it is difficult to compute <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(s)}" class="latex" /> exactly for most other values of <img src="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s}" class="latex" />.</p> <p>For each fixed <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" />, it is not hard to see that the expression <img src="https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\frac{1}{n^s} \eta(n/N) - C_{\eta,-s} N^{1-s}}" class="latex" /> is complex analytic in <img src="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s}" class="latex" />. Also, by a closer inspection of the error terms in the Euler-Maclaurin formula analysis, it is not difficult to show that for <img src="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s}" class="latex" /> in any compact region of <img src="https://s0.wp.com/latex.php?latex=%7B%5C%7B+s+%5Cin+%7B%5Cbf+C%7D%3A+%5Chbox%7BRe%7D%28s%29+%3C+1%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5C%7B+s+%5Cin+%7B%5Cbf+C%7D%3A+%5Chbox%7BRe%7D%28s%29+%3C+1%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5C%7B+s+%5Cin+%7B%5Cbf+C%7D%3A+%5Chbox%7BRe%7D%28s%29+%3C+1%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\{ s \in {\bf C}: \hbox{Re}(s) < 1\}}" class="latex" />, these expressions converge uniformly as <img src="https://s0.wp.com/latex.php?latex=%7BN+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N \rightarrow \infty}" class="latex" />. Applying <a href="http://en.wikipedia.org/wiki/Morera%27s_theorem">Morera’s theorem</a>, we conclude that our definition of <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(s)}" class="latex" /> is complex analytic in the region <img src="https://s0.wp.com/latex.php?latex=%7B%5C%7B+s+%5Cin+%7B%5Cbf+C%7D%3A+%5Chbox%7BRe%7D+s+%3C+1+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5C%7B+s+%5Cin+%7B%5Cbf+C%7D%3A+%5Chbox%7BRe%7D+s+%3C+1+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5C%7B+s+%5Cin+%7B%5Cbf+C%7D%3A+%5Chbox%7BRe%7D+s+%3C+1+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\{ s \in {\bf C}: \hbox{Re} s < 1 \}}" class="latex" />.</p> <p>We still have to connect this definition with the traditional definition <a href="#zeta-def">(1)</a> of the zeta function on the other half of the complex plane. To do this, we observe that</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D+%3D+%5Cint_0%5EN+x%5E%7B-s%7D+%5Ceta%28x%2FN%29%5C+dx+%3D+%5Cint_1%5EN+x%5E%7B-s%7D+%5Ceta%28x%2FN%29%5C+dx+-+%5Cfrac%7B1%7D%7Bs-1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D+%3D+%5Cint_0%5EN+x%5E%7B-s%7D+%5Ceta%28x%2FN%29%5C+dx+%3D+%5Cint_1%5EN+x%5E%7B-s%7D+%5Ceta%28x%2FN%29%5C+dx+-+%5Cfrac%7B1%7D%7Bs-1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D+%3D+%5Cint_0%5EN+x%5E%7B-s%7D+%5Ceta%28x%2FN%29%5C+dx+%3D+%5Cint_1%5EN+x%5E%7B-s%7D+%5Ceta%28x%2FN%29%5C+dx+-+%5Cfrac%7B1%7D%7Bs-1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle C_{\eta,-s} N^{1-s} = \int_0^N x^{-s} \eta(x/N)\ dx = \int_1^N x^{-s} \eta(x/N)\ dx - \frac{1}{s-1}" class="latex" /></p> <p>for <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" /> large enough. Thus we have</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29+%3D+%5Cfrac%7B1%7D%7Bs-1%7D+%2B+%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+%5Cint_1%5EN+x%5E%7B-s%7D+%5Ceta%28x%2FN%29%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29+%3D+%5Cfrac%7B1%7D%7Bs-1%7D+%2B+%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+%5Cint_1%5EN+x%5E%7B-s%7D+%5Ceta%28x%2FN%29%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29+%3D+%5Cfrac%7B1%7D%7Bs-1%7D+%2B+%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+-+%5Cint_1%5EN+x%5E%7B-s%7D+%5Ceta%28x%2FN%29%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \zeta(s) = \frac{1}{s-1} + \lim_{N \rightarrow \infty} \sum_{n=1}^\infty \frac{1}{n^s} \eta(n/N) - \int_1^N x^{-s} \eta(x/N)\ dx" class="latex" /></p> <p>for <img src="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D+s+%3C+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D+s+%3C+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D+s+%3C+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\hbox{Re} s < 1}" class="latex" />. The point of doing this is that this definition also makes sense in the region <img src="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\hbox{Re}(s) > 1}" class="latex" /> (due to the absolute convergence of the sum <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{n=1}^\infty \frac{1}{n^s}}" class="latex" /> and integral <img src="https://s0.wp.com/latex.php?latex=%7B%5Cint_1%5E%5Cinfty+x%5E%7B-s%7D+dx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cint_1%5E%5Cinfty+x%5E%7B-s%7D+dx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cint_1%5E%5Cinfty+x%5E%7B-s%7D+dx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\int_1^\infty x^{-s} dx}" class="latex" />. By using the trapezoidal rule, one also sees that this definition makes sense in the region <img src="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\hbox{Re}(s) > 0}" class="latex" />, with locally uniform convergence there also. So we in fact have a globally complex analytic definition of <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29+-+%5Cfrac%7B1%7D%7Bs-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29+-+%5Cfrac%7B1%7D%7Bs-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29+-+%5Cfrac%7B1%7D%7Bs-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(s) - \frac{1}{s-1}}" class="latex" />, and thus a <a href="http://en.wikipedia.org/wiki/Meromorphic_function">meromorphic</a> definition of <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(s)}" class="latex" /> on the complex plane. Note also that this definition gives the asymptotic <a name="zot"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29+%3D+%5Cfrac%7B1%7D%7Bs-1%7D+%2B+%5Cgamma+%2B+O%28%7Cs-1%7C%29+%5C+%5C+%5C+%5C+%5C+%2825%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29+%3D+%5Cfrac%7B1%7D%7Bs-1%7D+%2B+%5Cgamma+%2B+O%28%7Cs-1%7C%29+%5C+%5C+%5C+%5C+%5C+%2825%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29+%3D+%5Cfrac%7B1%7D%7Bs-1%7D+%2B+%5Cgamma+%2B+O%28%7Cs-1%7C%29+%5C+%5C+%5C+%5C+%5C+%2825%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \zeta(s) = \frac{1}{s-1} + \gamma + O(|s-1|) \ \ \ \ \ (25)" class="latex" /></p> <p><a name="zot"></a> near <img src="https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s=1}" class="latex" />, where <img src="https://s0.wp.com/latex.php?latex=%7B%5Cgamma%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cgamma%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cgamma%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\gamma}" class="latex" /> is <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant">Euler’s constant</a>.</p> <p>We have thus seen that asymptotics on smoothed partial sums of <img src="https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7B1%7D%7Bn%5Es%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7B1%7D%7Bn%5Es%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7B1%7D%7Bn%5Es%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\frac{1}{n^s}}" class="latex" /> gives rise to the familiar meromorphic properties of the Riemann zeta function <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(s)}" class="latex" />. It turns out that by combining the tools of Fourier analysis and complex analysis, one can reverse this procedure and deduce the asymptotics of <img src="https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7B1%7D%7Bn%5Es%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7B1%7D%7Bn%5Es%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7B1%7D%7Bn%5Es%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\frac{1}{n^s}}" class="latex" /> from the meromorphic properties of the zeta function.</p> <p>Let’s see how. Fix a complex number <img src="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29+%3C+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29+%3C+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29+%3C+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\hbox{Re}(s) < 1}" class="latex" />, and a smooth cutoff function <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%3A+%7B%5Cbf+R%7D%5E%2B+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%3A+%7B%5Cbf+R%7D%5E%2B+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%3A+%7B%5Cbf+R%7D%5E%2B+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta: {\bf R}^+ \rightarrow {\bf R}}" class="latex" /> which equals one near the origin, and consider the expression <a name="ins"></a></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+%5C+%5C+%5C+%5C+%5C+%2826%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+%5C+%5C+%5C+%5C+%5C+%2826%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+%5C+%5C+%5C+%5C+%5C+%2826%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \frac{1}{n^s} \eta(n/N) \ \ \ \ \ (26)" class="latex" /></p> <p><a name="ins"></a> where <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" /> is a large number. We let <img src="https://s0.wp.com/latex.php?latex=%7BA+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A > 0}" class="latex" /> be a large number, and rewrite this as</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+N%5E%7BA%7D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E%7Bs%2BA%7D%7D+f_A%28+%5Clog%28n%2FN%29+%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+N%5E%7BA%7D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E%7Bs%2BA%7D%7D+f_A%28+%5Clog%28n%2FN%29+%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+N%5E%7BA%7D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E%7Bs%2BA%7D%7D+f_A%28+%5Clog%28n%2FN%29+%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle N^{A} \sum_{n=1}^\infty \frac{1}{n^{s+A}} f_A( \log(n/N) )" class="latex" /></p> <p>where</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f_A%28x%29+%3A%3D+e%5E%7BAx%7D+%5Ceta%28+e%5Ex+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f_A%28x%29+%3A%3D+e%5E%7BAx%7D+%5Ceta%28+e%5Ex+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f_A%28x%29+%3A%3D+e%5E%7BAx%7D+%5Ceta%28+e%5Ex+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle f_A(x) := e^{Ax} \eta( e^x )." class="latex" /></p> <p>The function <img src="https://s0.wp.com/latex.php?latex=%7Bf_A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf_A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf_A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f_A}" class="latex" /> is in the Schwartz class. By the Fourier inversion formula, it has a Fourier representation</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f_A%28x%29+%3D+%5Cint_%7B%5Cbf+R%7D+%5Chat+f_A%28t%29+e%5E%7B-ixt%7D%5C+dt&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f_A%28x%29+%3D+%5Cint_%7B%5Cbf+R%7D+%5Chat+f_A%28t%29+e%5E%7B-ixt%7D%5C+dt&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f_A%28x%29+%3D+%5Cint_%7B%5Cbf+R%7D+%5Chat+f_A%28t%29+e%5E%7B-ixt%7D%5C+dt&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle f_A(x) = \int_{\bf R} \hat f_A(t) e^{-ixt}\ dt" class="latex" /></p> <p>where</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Chat+f_A%28x%29+%3A%3D+%5Cfrac%7B1%7D%7B2%5Cpi%7D+%5Cint_%7B%5Cbf+R%7D+f_A%28x%29+e%5E%7Bixt%7D%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Chat+f_A%28x%29+%3A%3D+%5Cfrac%7B1%7D%7B2%5Cpi%7D+%5Cint_%7B%5Cbf+R%7D+f_A%28x%29+e%5E%7Bixt%7D%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Chat+f_A%28x%29+%3A%3D+%5Cfrac%7B1%7D%7B2%5Cpi%7D+%5Cint_%7B%5Cbf+R%7D+f_A%28x%29+e%5E%7Bixt%7D%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \hat f_A(x) := \frac{1}{2\pi} \int_{\bf R} f_A(x) e^{ixt}\ dx" class="latex" /></p> <p>and so <a href="#ins">(26)</a> can be rewritten as</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+N%5E%7BA%7D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E%7Bs%2BA%7D%7D+%5Cint_%7B%5Cbf+R%7D+%5Chat+f_A%28t%29+%28n%2FN%29%5E%7B-it%7D%5C+dt.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+N%5E%7BA%7D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E%7Bs%2BA%7D%7D+%5Cint_%7B%5Cbf+R%7D+%5Chat+f_A%28t%29+%28n%2FN%29%5E%7B-it%7D%5C+dt.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+N%5E%7BA%7D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E%7Bs%2BA%7D%7D+%5Cint_%7B%5Cbf+R%7D+%5Chat+f_A%28t%29+%28n%2FN%29%5E%7B-it%7D%5C+dt.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle N^{A} \sum_{n=1}^\infty \frac{1}{n^{s+A}} \int_{\bf R} \hat f_A(t) (n/N)^{-it}\ dt." class="latex" /></p> <p>The function <img src="https://s0.wp.com/latex.php?latex=%7B%5Chat+f_A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Chat+f_A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Chat+f_A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\hat f_A}" class="latex" /> is also Schwartz. If <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> is large enough, we may then interchange the integral and sum and use <a href="#zeta-def">(1)</a> to rewrite <a href="#ins">(26)</a> as</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%5Cbf+R%7D+%5Chat+f_A%28t%29+N%5E%7BA%2Bit%7D+%5Czeta%28s%2BA%2Bit%29%5C+dt.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%5Cbf+R%7D+%5Chat+f_A%28t%29+N%5E%7BA%2Bit%7D+%5Czeta%28s%2BA%2Bit%29%5C+dt.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%5Cbf+R%7D+%5Chat+f_A%28t%29+N%5E%7BA%2Bit%7D+%5Czeta%28s%2BA%2Bit%29%5C+dt.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_{\bf R} \hat f_A(t) N^{A+it} \zeta(s+A+it)\ dt." class="latex" /></p> <p>Now we have</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Chat+f_A%28t%29+%3D+%5Cfrac%7B1%7D%7B2%5Cpi%7D+%5Cint_%7B%5Cbf+R%7D+e%5E%7B%28A%2Bit%29x%7D+%5Ceta%28e%5Ex%29%5C+dx%3B&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Chat+f_A%28t%29+%3D+%5Cfrac%7B1%7D%7B2%5Cpi%7D+%5Cint_%7B%5Cbf+R%7D+e%5E%7B%28A%2Bit%29x%7D+%5Ceta%28e%5Ex%29%5C+dx%3B&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Chat+f_A%28t%29+%3D+%5Cfrac%7B1%7D%7B2%5Cpi%7D+%5Cint_%7B%5Cbf+R%7D+e%5E%7B%28A%2Bit%29x%7D+%5Ceta%28e%5Ex%29%5C+dx%3B&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \hat f_A(t) = \frac{1}{2\pi} \int_{\bf R} e^{(A+it)x} \eta(e^x)\ dx;" class="latex" /></p> <p>integrating by parts (which is justified when <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> is large enough) we have</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Chat+f_A%28t%29+%3D+-%5Cfrac%7B1%7D%7B2%5Cpi+%28A%2Bit%29%7D+F%28A%2Bit%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Chat+f_A%28t%29+%3D+-%5Cfrac%7B1%7D%7B2%5Cpi+%28A%2Bit%29%7D+F%28A%2Bit%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Chat+f_A%28t%29+%3D+-%5Cfrac%7B1%7D%7B2%5Cpi+%28A%2Bit%29%7D+F%28A%2Bit%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \hat f_A(t) = -\frac{1}{2\pi (A+it)} F(A+it)" class="latex" /></p> <p>where</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+F%28A%2Bit%29+%3D+%5Cint_%7B%5Cbf+R%7D+e%5E%7B%28A%2Bit%2B1%29x%7D+%5Ceta%27%28e%5Ex%29%5C+dx+%3D+%5Cint_0%5E%5Cinfty+y%5E%7BA%2Bit%7D+%5Ceta%27%28y%29%5C+dy.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+F%28A%2Bit%29+%3D+%5Cint_%7B%5Cbf+R%7D+e%5E%7B%28A%2Bit%2B1%29x%7D+%5Ceta%27%28e%5Ex%29%5C+dx+%3D+%5Cint_0%5E%5Cinfty+y%5E%7BA%2Bit%7D+%5Ceta%27%28y%29%5C+dy.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+F%28A%2Bit%29+%3D+%5Cint_%7B%5Cbf+R%7D+e%5E%7B%28A%2Bit%2B1%29x%7D+%5Ceta%27%28e%5Ex%29%5C+dx+%3D+%5Cint_0%5E%5Cinfty+y%5E%7BA%2Bit%7D+%5Ceta%27%28y%29%5C+dy.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle F(A+it) = \int_{\bf R} e^{(A+it+1)x} \eta'(e^x)\ dx = \int_0^\infty y^{A+it} \eta'(y)\ dy." class="latex" /></p> <p>We can thus write <a href="#ins">(26)</a> as a contour integral</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B-1%7D%7B2%5Cpi+i%7D+%5Cint_%7Bs%2BA-i%5Cinfty%7D%5E%7Bs%2BA%2Bi%5Cinfty%7D+%5Czeta%28z%29+%5Cfrac%7BN%5E%7Bz-s%7D+F%28z-s%29%7D%7Bz-s%7D%5C+dz.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B-1%7D%7B2%5Cpi+i%7D+%5Cint_%7Bs%2BA-i%5Cinfty%7D%5E%7Bs%2BA%2Bi%5Cinfty%7D+%5Czeta%28z%29+%5Cfrac%7BN%5E%7Bz-s%7D+F%28z-s%29%7D%7Bz-s%7D%5C+dz.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B-1%7D%7B2%5Cpi+i%7D+%5Cint_%7Bs%2BA-i%5Cinfty%7D%5E%7Bs%2BA%2Bi%5Cinfty%7D+%5Czeta%28z%29+%5Cfrac%7BN%5E%7Bz-s%7D+F%28z-s%29%7D%7Bz-s%7D%5C+dz.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{-1}{2\pi i} \int_{s+A-i\infty}^{s+A+i\infty} \zeta(z) \frac{N^{z-s} F(z-s)}{z-s}\ dz." class="latex" /></p> <p>Note that <img src="https://s0.wp.com/latex.php?latex=%7B%5Ceta%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ceta%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ceta%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\eta'}" class="latex" /> is compactly supported away from zero, which makes <img src="https://s0.wp.com/latex.php?latex=%7BF%28A%2Bit%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%28A%2Bit%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%28A%2Bit%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F(A+it)}" class="latex" /> an entire function of <img src="https://s0.wp.com/latex.php?latex=%7BA%2Bit%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2Bit%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2Bit%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A+it}" class="latex" />, which is uniformly bounded whenever <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> is bounded. Furthermore, from repeated integration by parts we see that <img src="https://s0.wp.com/latex.php?latex=%7BF%28A%2Bit%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%28A%2Bit%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%28A%2Bit%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F(A+it)}" class="latex" /> is rapidly decreasing as <img src="https://s0.wp.com/latex.php?latex=%7Bt+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t \rightarrow \infty}" class="latex" />, uniformly for <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> in a compact set. Meanwhile, standard estimates show that <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28%5Csigma%2Bit%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28%5Csigma%2Bit%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28%5Csigma%2Bit%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(\sigma+it)}" class="latex" /> is of polynomial growth in <img src="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t}" class="latex" /> for <img src="https://s0.wp.com/latex.php?latex=%7B%5Csigma%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csigma%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csigma%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sigma}" class="latex" /> in a compact set. Finally, the meromorphic function <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28z%29+%5Cfrac%7BN%5E%7Bz-s%7D+F%28z-s%29%7D%7Bz-s%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28z%29+%5Cfrac%7BN%5E%7Bz-s%7D+F%28z-s%29%7D%7Bz-s%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28z%29+%5Cfrac%7BN%5E%7Bz-s%7D+F%28z-s%29%7D%7Bz-s%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(z) \frac{N^{z-s} F(z-s)}{z-s}}" class="latex" /> has a simple pole at <img src="https://s0.wp.com/latex.php?latex=%7Bz%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bz%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bz%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{z=1}" class="latex" /> (with residue <img src="https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7BN%5E%7B1-s%7D+F%281-s%29%7D%7B1-s%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7BN%5E%7B1-s%7D+F%281-s%29%7D%7B1-s%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7BN%5E%7B1-s%7D+F%281-s%29%7D%7B1-s%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\frac{N^{1-s} F(1-s)}{1-s}}" class="latex" />) and at <img src="https://s0.wp.com/latex.php?latex=%7Bz-s%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bz-s%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bz-s%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{z-s}" class="latex" /> (with residue <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29+F%280%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29+F%280%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29+F%280%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(s) F(0)}" class="latex" />). Applying the residue theorem, we can write <a href="#ins">(26)</a> as</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B-1%7D%7B2%5Cpi+i%7D+%5Cint_%7Bs-B-i%5Cinfty%7D%5E%7Bs-B%2Bi%5Cinfty%7D+%5Czeta%28z%29+%5Cfrac%7BN%5E%7Bz-s%7D+F%28z-s%29%7D%7Bz-s%7D%5C+dz+-+%5Cfrac%7BN%5E%7B1-s%7D+F%281-s%29%7D%7B1-s%7D+-+%5Czeta%28s%29+F%280%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B-1%7D%7B2%5Cpi+i%7D+%5Cint_%7Bs-B-i%5Cinfty%7D%5E%7Bs-B%2Bi%5Cinfty%7D+%5Czeta%28z%29+%5Cfrac%7BN%5E%7Bz-s%7D+F%28z-s%29%7D%7Bz-s%7D%5C+dz+-+%5Cfrac%7BN%5E%7B1-s%7D+F%281-s%29%7D%7B1-s%7D+-+%5Czeta%28s%29+F%280%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B-1%7D%7B2%5Cpi+i%7D+%5Cint_%7Bs-B-i%5Cinfty%7D%5E%7Bs-B%2Bi%5Cinfty%7D+%5Czeta%28z%29+%5Cfrac%7BN%5E%7Bz-s%7D+F%28z-s%29%7D%7Bz-s%7D%5C+dz+-+%5Cfrac%7BN%5E%7B1-s%7D+F%281-s%29%7D%7B1-s%7D+-+%5Czeta%28s%29+F%280%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{-1}{2\pi i} \int_{s-B-i\infty}^{s-B+i\infty} \zeta(z) \frac{N^{z-s} F(z-s)}{z-s}\ dz - \frac{N^{1-s} F(1-s)}{1-s} - \zeta(s) F(0)" class="latex" /></p> <p>for any <img src="https://s0.wp.com/latex.php?latex=%7BB%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B>0}" class="latex" />. Using the various bounds on <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F}" class="latex" />, we see that the integral is <img src="https://s0.wp.com/latex.php?latex=%7BO%28+N%5E%7B-B%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28+N%5E%7B-B%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28+N%5E%7B-B%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O( N^{-B})}" class="latex" />. From integration by parts we have <img src="https://s0.wp.com/latex.php?latex=%7BF%280%29%3D-1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%280%29%3D-1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%280%29%3D-1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F(0)=-1}" class="latex" /> and</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+F%281-s%29+%3D+-+%281-s%29+%5Cint_0%5E%5Cinfty+y%5E%7B-s%7D+%5Ceta%28y%29%5C+dy+%3D+-%281-s%29+C_%7B%5Ceta%2C-s%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+F%281-s%29+%3D+-+%281-s%29+%5Cint_0%5E%5Cinfty+y%5E%7B-s%7D+%5Ceta%28y%29%5C+dy+%3D+-%281-s%29+C_%7B%5Ceta%2C-s%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+F%281-s%29+%3D+-+%281-s%29+%5Cint_0%5E%5Cinfty+y%5E%7B-s%7D+%5Ceta%28y%29%5C+dy+%3D+-%281-s%29+C_%7B%5Ceta%2C-s%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle F(1-s) = - (1-s) \int_0^\infty y^{-s} \eta(y)\ dy = -(1-s) C_{\eta,-s}" class="latex" /></p> <p>and thus we have</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+%3D+%5Czeta%28s%29+%2B+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D+%2B+O%28N%5E%7B-B%7D%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+%3D+%5Czeta%28s%29+%2B+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D+%2B+O%28N%5E%7B-B%7D%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+%3D+%5Czeta%28s%29+%2B+C_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D+%2B+O%28N%5E%7B-B%7D%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \frac{1}{n^s} \eta(n/N) = \zeta(s) + C_{\eta,-s} N^{1-s} + O(N^{-B})" class="latex" /></p> <p>for any <img src="https://s0.wp.com/latex.php?latex=%7BB%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B>0}" class="latex" />, which is <a href="#zeta-asym-s">(14)</a> (with the refined error term indicated in Remark <a href="#nab">4</a>).</p> <p>The above argument reveals that the simple pole of <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta(s)}" class="latex" /> at <img src="https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s=1}" class="latex" /> is directly connected to the <img src="https://s0.wp.com/latex.php?latex=%7BC_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BC_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BC_%7B%5Ceta%2C-s%7D+N%5E%7B1-s%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{C_{\eta,-s} N^{1-s}}" class="latex" /> term in the asymptotics of the smoothed partial sums. More generally, if a Dirichlet series</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+D%28s%29+%3D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7Ba_n%7D%7Bn%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+D%28s%29+%3D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7Ba_n%7D%7Bn%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+D%28s%29+%3D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7Ba_n%7D%7Bn%5Es%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle D(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}" class="latex" /></p> <p>has a meromorphic continuation to the entire complex plane, and does not grow too fast at infinity, then one (heuristically at least) has the asymptotic</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7Ba_n%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+%3D+D%28s%29+%2B+%5Csum_%5Crho+C_%7B%5Ceta%2C%5Crho-s-1%7D+r_%5Crho+N%5E%7B%5Crho-s%7D+%2B+%5Cldots&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7Ba_n%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+%3D+D%28s%29+%2B+%5Csum_%5Crho+C_%7B%5Ceta%2C%5Crho-s-1%7D+r_%5Crho+N%5E%7B%5Crho-s%7D+%2B+%5Cldots&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7Ba_n%7D%7Bn%5Es%7D+%5Ceta%28n%2FN%29+%3D+D%28s%29+%2B+%5Csum_%5Crho+C_%7B%5Ceta%2C%5Crho-s-1%7D+r_%5Crho+N%5E%7B%5Crho-s%7D+%2B+%5Cldots&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \frac{a_n}{n^s} \eta(n/N) = D(s) + \sum_\rho C_{\eta,\rho-s-1} r_\rho N^{\rho-s} + \ldots" class="latex" /></p> <p>where <img src="https://s0.wp.com/latex.php?latex=%7B%5Crho%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Crho%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Crho%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\rho}" class="latex" /> ranges over the poles of <img src="https://s0.wp.com/latex.php?latex=%7BD%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BD%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BD%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{D}" class="latex" />, and <img src="https://s0.wp.com/latex.php?latex=%7Br_%5Crho%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Br_%5Crho%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Br_%5Crho%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{r_\rho}" class="latex" /> are the residues at those poles. For instance, one has the famous <em>explicit formula</em></p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5CLambda%28n%29+%5Ceta%28n%2FN%29+%3D+C_%7B%5Ceta%2C0%7D+N+-+%5Csum_%5Crho+C_%7B%5Ceta%2C%5Crho-1%7D+N%5E%5Crho+%2B+%5Cldots&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5CLambda%28n%29+%5Ceta%28n%2FN%29+%3D+C_%7B%5Ceta%2C0%7D+N+-+%5Csum_%5Crho+C_%7B%5Ceta%2C%5Crho-1%7D+N%5E%5Crho+%2B+%5Cldots&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5CLambda%28n%29+%5Ceta%28n%2FN%29+%3D+C_%7B%5Ceta%2C0%7D+N+-+%5Csum_%5Crho+C_%7B%5Ceta%2C%5Crho-1%7D+N%5E%5Crho+%2B+%5Cldots&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \Lambda(n) \eta(n/N) = C_{\eta,0} N - \sum_\rho C_{\eta,\rho-1} N^\rho + \ldots" class="latex" /></p> <p>where <img src="https://s0.wp.com/latex.php?latex=%7B%5CLambda%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CLambda%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CLambda%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Lambda}" class="latex" /> is the <a href="http://en.wikipedia.org/wiki/Von_Mangoldt_function">von Mangoldt function</a>, <img src="https://s0.wp.com/latex.php?latex=%7B%5Crho%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Crho%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Crho%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\rho}" class="latex" /> are the non-trivial zeroes of the Riemann zeta function (counting multiplicity, if any), and <img src="https://s0.wp.com/latex.php?latex=%7B%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cldots%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\ldots}" class="latex" /> is an error term (basically arising from the trivial zeroes of zeta); this ultimately reflects the fact that the Dirichlet series</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B%5CLambda%28n%29%7D%7Bn%5Es%7D+%3D+-%5Cfrac%7B%5Czeta%27%28s%29%7D%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B%5CLambda%28n%29%7D%7Bn%5Es%7D+%3D+-%5Cfrac%7B%5Czeta%27%28s%29%7D%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B%5CLambda%28n%29%7D%7Bn%5Es%7D+%3D+-%5Cfrac%7B%5Czeta%27%28s%29%7D%7B%5Czeta%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s} = -\frac{\zeta'(s)}{\zeta(s)}" class="latex" /></p> <p>has a simple pole at <img src="https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s=1}" class="latex" /> (with residue <img src="https://s0.wp.com/latex.php?latex=%7B%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{+1}" class="latex" />) and simple poles at every zero of the zeta function with residue <img src="https://s0.wp.com/latex.php?latex=%7B-1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B-1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B-1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{-1}" class="latex" /> (weighted again by multiplicity, though it is not believed that multiple zeroes actually exist).</p> <p>The link between poles of the zeta function (and its relatives) and asymptotics of (smoothed) partial sums of arithmetical functions can be used to compare elementary methods in analytic number theory with complex methods. Roughly speaking, elementary methods are based on leading term asymptotics of partial sums of arithmetical functions, and are mostly based on exploiting the simple pole of <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta}" class="latex" /> at <img src="https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s=1}" class="latex" /> (and the lack of a simple zero of Dirichlet <img src="https://s0.wp.com/latex.php?latex=%7BL%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BL%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BL%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{L}" class="latex" />-functions at <img src="https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s=1}" class="latex" />); in contrast, complex methods also take full advantage of the zeroes of <img src="https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Czeta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\zeta}" class="latex" /> and Dirichlet <img src="https://s0.wp.com/latex.php?latex=%7BL%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BL%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BL%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{L}" class="latex" />-functions (or the lack thereof) in the entire complex plane, as well as the functional equation (which, in terms of smoothed partial sums, manifests itself through the <a href="http://en.wikipedia.org/wiki/Poisson_summation_formula">Poisson summation formula</a>). Indeed, using the above correspondences it is not hard to see that the prime number theorem (for instance) is equivalent to the lack of zeroes of the Riemann zeta function on the line <img src="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Chbox%7BRe%7D%28s%29%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\hbox{Re}(s)=1}" class="latex" />.</p> <p>With this dictionary between elementary methods and complex methods, the Dirichlet hyperbola method in elementary analytic number theory corresponds to analysing the behaviour of poles and residues when multiplying together two Dirichlet series. For instance, by using the formula <a href="#zeta-asym-1">(11)</a> and the hyperbola method, together with the asymptotic</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%7D+%5Ceta%28n%2FN%29+%3D+%5Cint_1%5E%5Cinfty+%5Ceta%28x%2FN%29%5C+%5Cfrac%7Bdx%7D%7Bx%7D+%2B+%5Cgamma+%2B+O%281%2FN%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%7D+%5Ceta%28n%2FN%29+%3D+%5Cint_1%5E%5Cinfty+%5Ceta%28x%2FN%29%5C+%5Cfrac%7Bdx%7D%7Bx%7D+%2B+%5Cgamma+%2B+O%281%2FN%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%7D+%5Ceta%28n%2FN%29+%3D+%5Cint_1%5E%5Cinfty+%5Ceta%28x%2FN%29%5C+%5Cfrac%7Bdx%7D%7Bx%7D+%2B+%5Cgamma+%2B+O%281%2FN%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \frac{1}{n} \eta(n/N) = \int_1^\infty \eta(x/N)\ \frac{dx}{x} + \gamma + O(1/N)" class="latex" /></p> <p>which can be obtained from the trapezoidal rule and the definition of <img src="https://s0.wp.com/latex.php?latex=%7B%5Cgamma%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cgamma%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cgamma%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\gamma}" class="latex" />, one can obtain the asymptotic</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ctau%28n%29+%5Ceta%28n%2FN%29+%3D+%5Cint_1%5E%5Cinfty+%5Clog+x+%5Ceta%28x%2FN%29%5C+dx+%2B+2%5Cgamma+C_%7B%5Ceta%2C0%7D+N+%2B+O%28%5Csqrt%7BN%7D%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ctau%28n%29+%5Ceta%28n%2FN%29+%3D+%5Cint_1%5E%5Cinfty+%5Clog+x+%5Ceta%28x%2FN%29%5C+dx+%2B+2%5Cgamma+C_%7B%5Ceta%2C0%7D+N+%2B+O%28%5Csqrt%7BN%7D%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Ctau%28n%29+%5Ceta%28n%2FN%29+%3D+%5Cint_1%5E%5Cinfty+%5Clog+x+%5Ceta%28x%2FN%29%5C+dx+%2B+2%5Cgamma+C_%7B%5Ceta%2C0%7D+N+%2B+O%28%5Csqrt%7BN%7D%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \tau(n) \eta(n/N) = \int_1^\infty \log x \eta(x/N)\ dx + 2\gamma C_{\eta,0} N + O(\sqrt{N})" class="latex" /></p> <p>where <img src="https://s0.wp.com/latex.php?latex=%7B%5Ctau%28n%29+%3A%3D+%5Csum_%7Bd%7Cn%7D+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ctau%28n%29+%3A%3D+%5Csum_%7Bd%7Cn%7D+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ctau%28n%29+%3A%3D+%5Csum_%7Bd%7Cn%7D+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\tau(n) := \sum_{d|n} 1}" class="latex" /> is the divisor function (and in fact one can improve the <img src="https://s0.wp.com/latex.php?latex=%7BO%28%5Csqrt%7BN%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28%5Csqrt%7BN%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28%5Csqrt%7BN%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(\sqrt{N})}" class="latex" /> bound substantially by being more careful); this corresponds to the fact that the Dirichlet series</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B%5Ctau%28n%29%7D%7Bn%5Es%7D+%3D+%5Czeta%28s%29%5E2&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B%5Ctau%28n%29%7D%7Bn%5Es%7D+%3D+%5Czeta%28s%29%5E2&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B%5Ctau%28n%29%7D%7Bn%5Es%7D+%3D+%5Czeta%28s%29%5E2&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \frac{\tau(n)}{n^s} = \zeta(s)^2" class="latex" /></p> <p>has a double pole at <img src="https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s=1}" class="latex" /> with expansion</p> <p align="center"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29%5E2+%3D+%5Cfrac%7B1%7D%7B%28s-1%29%5E2%7D+%2B+2+%5Cgamma+%5Cfrac%7B1%7D%7Bs-1%7D+%2B+O%281%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29%5E2+%3D+%5Cfrac%7B1%7D%7B%28s-1%29%5E2%7D+%2B+2+%5Cgamma+%5Cfrac%7B1%7D%7Bs-1%7D+%2B+O%281%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Czeta%28s%29%5E2+%3D+%5Cfrac%7B1%7D%7B%28s-1%29%5E2%7D+%2B+2+%5Cgamma+%5Cfrac%7B1%7D%7Bs-1%7D+%2B+O%281%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \zeta(s)^2 = \frac{1}{(s-1)^2} + 2 \gamma \frac{1}{s-1} + O(1)" class="latex" /></p> <p>and no other poles, which of course follows by multiplying <a href="#zot">(25)</a> with itself.</p> <blockquote><p><b>Remark 6</b> In the literature, elementary methods in analytic number theorem often use sharply truncated sums rather than smoothed sums. However, as indicated earlier, the error terms tend to be slightly better when working with smoothed sums (although not much gain is obtained in this manner when dealing with sums of functions that are sensitive to the primes, such as <img src="https://s0.wp.com/latex.php?latex=%7B%5CLambda%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CLambda%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CLambda%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Lambda}" class="latex" />, as the terms arising from the zeroes of the zeta function tend to dominate any saving in this regard).</p></blockquote> <div id="jp-post-flair" class="sharedaddy sd-like-enabled sd-sharing-enabled"><div class="sharedaddy sd-sharing-enabled"><div class="robots-nocontent sd-block sd-social sd-social-icon-text sd-sharing"><h3 class="sd-title">Share this:</h3><div class="sd-content"><ul><li class="share-print"><a rel="nofollow noopener noreferrer" data-shared="" class="share-print sd-button share-icon" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/#print" target="_blank" title="Click to print" ><span>Print</span></a></li><li class="share-email"><a rel="nofollow noopener noreferrer" data-shared="" class="share-email sd-button share-icon" href="mailto:?subject=%5BShared%20Post%5D%20The%20Euler-Maclaurin%20formula%2C%20Bernoulli%20numbers%2C%20the%20zeta%20function%2C%20and%20real-variable%20analytic%20continuation&body=https%3A%2F%2Fterrytao.wordpress.com%2F2010%2F04%2F10%2Fthe-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation%2F&share=email" target="_blank" title="Click to email a link to a friend" data-email-share-error-title="Do you have email set up?" data-email-share-error-text="If you're having problems sharing via email, you might not have email set up for your browser. 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width:32px;"><img alt='' src='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /></td><td class="recentcommentstextend" style="">Anonymous on <a href="https://terrytao.wordpress.com/2025/02/13/cosmic-distance-ladder-video-with-grant-sanderson-3blue1brown-commentary-and-corrections/comment-page-1/#comment-687194">Cosmic Distance Ladder video w…</a></td></tr><tr><td title="" class="recentcommentsavatarend" style="height:32px; width:32px;"><img alt='' src='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /></td><td class="recentcommentstextend" style="">Anonymous on <a href="https://terrytao.wordpress.com/2025/02/13/cosmic-distance-ladder-video-with-grant-sanderson-3blue1brown-commentary-and-corrections/comment-page-1/#comment-687193">Cosmic Distance Ladder video w…</a></td></tr><tr><td title="Terence Tao" class="recentcommentsavatarend" style="height:32px; width:32px;"><a href="http://www.math.ucla.edu/~tao" rel="nofollow"><img alt='' src='https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=32&d=identicon&r=PG' srcset='https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=32&d=identicon&r=PG 1x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=48&d=identicon&r=PG 1.5x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=64&d=identicon&r=PG 2x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=96&d=identicon&r=PG 3x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /></a></td><td class="recentcommentstextend" style=""><a href="http://www.math.ucla.edu/~tao" rel="nofollow">Terence Tao</a> on <a href="https://terrytao.wordpress.com/2025/02/13/cosmic-distance-ladder-video-with-grant-sanderson-3blue1brown-commentary-and-corrections/comment-page-1/#comment-687192">Cosmic Distance Ladder video w…</a></td></tr><tr><td title="Aditya Guha Roy" class="recentcommentsavatarend" style="height:32px; 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width:32px;"><img alt='' src='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /></td><td class="recentcommentstextend" style="">Anonymous on <a href="https://terrytao.wordpress.com/2025/02/13/cosmic-distance-ladder-video-with-grant-sanderson-3blue1brown-commentary-and-corrections/comment-page-1/#comment-687190">Cosmic Distance Ladder video w…</a></td></tr><tr><td title="Terence Tao" class="recentcommentsavatarend" style="height:32px; width:32px;"><a href="http://www.math.ucla.edu/~tao" rel="nofollow"><img alt='' src='https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=32&d=identicon&r=PG' srcset='https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=32&d=identicon&r=PG 1x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=48&d=identicon&r=PG 1.5x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=64&d=identicon&r=PG 2x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=96&d=identicon&r=PG 3x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /></a></td><td class="recentcommentstextend" style=""><a href="http://www.math.ucla.edu/~tao" rel="nofollow">Terence Tao</a> on <a href="https://terrytao.wordpress.com/2025/02/13/cosmic-distance-ladder-video-with-grant-sanderson-3blue1brown-commentary-and-corrections/comment-page-1/#comment-687189">Cosmic Distance Ladder video w…</a></td></tr><tr><td title="" class="recentcommentsavatarend" style="height:32px; width:32px;"><img alt='' src='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /></td><td class="recentcommentstextend" style="">Anonymous on <a href="https://terrytao.wordpress.com/2025/02/13/cosmic-distance-ladder-video-with-grant-sanderson-3blue1brown-commentary-and-corrections/comment-page-1/#comment-687188">Cosmic Distance Ladder video w…</a></td></tr><tr><td title="" class="recentcommentsavatarend" style="height:32px; width:32px;"><img alt='' src='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /></td><td class="recentcommentstextend" style="">Anonymous on <a href="https://terrytao.wordpress.com/2025/02/13/cosmic-distance-ladder-video-with-grant-sanderson-3blue1brown-commentary-and-corrections/comment-page-1/#comment-687187">Cosmic Distance Ladder video w…</a></td></tr><tr><td title="Sam" class="recentcommentsavatarend" style="height:32px; width:32px;"><a href="http://yangshu220.wordpress.com" rel="nofollow"><img alt='' src='https://1.gravatar.com/avatar/a96b6b5273499d10b7c792cf597c64511feb6b2360aea19c9d72565c83688692?s=32&d=identicon&r=PG' srcset='https://1.gravatar.com/avatar/a96b6b5273499d10b7c792cf597c64511feb6b2360aea19c9d72565c83688692?s=32&d=identicon&r=PG 1x, https://1.gravatar.com/avatar/a96b6b5273499d10b7c792cf597c64511feb6b2360aea19c9d72565c83688692?s=48&d=identicon&r=PG 1.5x, https://1.gravatar.com/avatar/a96b6b5273499d10b7c792cf597c64511feb6b2360aea19c9d72565c83688692?s=64&d=identicon&r=PG 2x, https://1.gravatar.com/avatar/a96b6b5273499d10b7c792cf597c64511feb6b2360aea19c9d72565c83688692?s=96&d=identicon&r=PG 3x, https://1.gravatar.com/avatar/a96b6b5273499d10b7c792cf597c64511feb6b2360aea19c9d72565c83688692?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /></a></td><td class="recentcommentstextend" style=""><a href="http://yangshu220.wordpress.com" rel="nofollow">Sam</a> on <a href="https://terrytao.wordpress.com/2015/11/02/275a-notes-4-the-central-limit-theorem/comment-page-2/#comment-687186">275A, Notes 4: The central lim…</a></td></tr><tr><td title="" class="recentcommentsavatarend" style="height:32px; 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Every Tuesday.">What if? (xkcd)</a></li> <li><a href="http://www.xkcd.com/" title="A webcomic of romance, sarcasm, math, and language">xkcd</a></li> </ul> </div> <div id="linkcat-3582" class="widget widget_links"><h3>Mathematics</h3> <ul class='xoxo blogroll'> <li><a href="https://11011110.github.io/blog/" title="David Eppstein’s blog">0xDE</a></li> <li><a href="http://hilbertthm90.wordpress.com/" title="Musings on art, philosophy, mathematics, and physics">A Mind for Madness</a></li> <li><a href="http://mzargar.wordpress.com/" title="Olympiad level mathematical problems">A Portion of the Book</a></li> <li><a href="http://sunejakobsen.wordpress.com/">Absolutely useless</a></li> <li><a href="http://alexsisto.wordpress.com/" title="Alessandro Sisto’s math blog">Alex Sisto</a></li> <li><a href="https://algorithmsoup.wordpress.com/" title="A bit of math and a byte of computer science">Algorithm Soup</a></li> <li><a href="https://almostoriginality.wordpress.com/" title="a mathematical journal">Almost Originality</a></li> <li><a href="http://blogs.ams.org/" title="Blogs offered by or in cooperation with the American Mathematical Society">AMS blogs</a></li> <li><a href="http://mathgradblog.williams.edu/" title="by and for math grad students">AMS Graduate Student Blog</a></li> <li><a href="http://pjm.math.berkeley.edu/apde/about/journal/about.html" title="A new journal in analysis & PDE (I am one of the editors), run by and for mathematicians">Analysis & PDE</a></li> <li><a href="https://kvm16.pims.math.ca/DispersiveWiki/index.php?title=Conferences" title="A list of upcoming conferences in analysis & PDE">Analysis & PDE Conferences</a></li> <li><a href="http://qchu.wordpress.com/" title="Mathematicians are annoyingly precise.">Annoying Precision</a></li> <li><a href="http://conan777.wordpress.com/" title="Conan Wu’s mathematical blog">Area 777</a></li> <li><a href="http://www.arsmathematica.net/" title="Dedicated to the mathematical arts.">Ars Mathematica</a></li> <li><a href="http://brauer.maths.qmul.ac.uk/Atlas/v3/">ATLAS of Finite Group Representations</a></li> <li><a href="http://automorphicforum.wordpress.com/" title="Automorphic forms, number theory, representation theory, algebraic geometry, and other mathematics">Automorphic forum</a></li> <li><a href="http://avzel.blogspot.com/" title="Andrei Zelevinsky’s blog (in Russian)">Avzel's journal</a></li> <li><a href="http://blogs.ams.org/blogonmathblogs/" title="Two mathematicians tour the mathematical blogosphere (hosted by the American Mathematical Society)">Blog on Math Blogs</a></li> <li><a href="https://blogderbeweise.wordpress.com" title="This one is from the book !">blogderbeweise</a></li> <li><a href="http://williewong.wordpress.com/">Bubbles Bad; Ripples Good</a></li> <li><a href="http://cedricvillani.org/">Cédric Villani</a></li> <li><a href="http://amathew.wordpress.com/" title="Thoughts on mathematics">Climbing Mount Bourbaki</a></li> <li><a href="http://coloquiooleis.wordpress.com/" title="los seguidores de Manolo (Spanish)">Coloquio Oleis</a></li> <li><a href="http://gilkalai.wordpress.com/" rel="friend colleague" title="Gil Kalai’s blog">Combinatorics and more</a></li> <li><a href="http://www.compressedsensing.com/" title="Maintained by Rice university">Compressed sensing resources</a></li> <li><a href="http://weblog.fortnow.com/" title="by Lance Fortnow and Bill Gasarch">Computational Complexity</a></li> <li><a href="http://concretenonsense.wordpress.com/" title="A group blog about mathematics">Concrete nonsense</a></li> <li><a href="http://www.dam.brown.edu/people/mumford/blog.html">David Mumford's blog</a></li> <li><a href="http://deltaepsilons.wordpress.com/" title="Mathematical research and problem solving">Delta epsilons</a></li> <li><a href="https://kvm16.pims.math.ca/DispersiveWiki/index.php?title=Main_Page" title="A wiki for nonlinear dispersive and wave equations">DispersiveWiki</a></li> <li><a href="http://matheuscmss.wordpress.com/" title="Carlos Matheus’ blog">Disquisitiones Mathematicae</a></li> <li><a href="http://embuchestissues.wordpress.com/">Embûches tissues</a></li> <li><a href="http://blogs.ethz.ch/kowalski/" title="Emmanuel Kowalski’s blog">Emmanuel Kowalski’s blog</a></li> <li><a href="http://equatorialmaths.wordpress.com/" title="Mathematics, teaching, brain meltdown">Equatorial Mathematics</a></li> <li><a href="https://www.erdosproblems.com/" title="A database of Erdos problems and their current status">Erdos problems</a></li> <li><a href="https://ffbandf.wordpress.com/" title="Forking, Forcing and back&Forthing">fff</a></li> <li><a href="http://floerhomology.wordpress.com/" title="Michael Hutchings’ blog">Floer Homology</a></li> <li><a href="http://blogs.williams.edu/Morgan/" title="Math, Teaching, and Other Items of Interest">Frank Morgan’s blog</a></li> <li><a href="http://gerardbesson.wordpress.com/">Gérard Besson's Blog</a></li> <li><a href="http://rjlipton.wordpress.com/" title="http://rjlipton.wordpress.com/">Gödel’s Lost Letter and P=NP</a></li> <li><a href="http://392c.wordpress.com/" title="Residual finiteness and word-hyperbolic groups">Geometric Group Theory</a></li> <li><a href="http://lamington.wordpress.com/" title="Geometry, explored in examples; and commentary on recent developments (by Danny Calegari)">Geometry and the imagination</a></li> <li><a href="http://burttotaro.wordpress.com">Geometry Bulletin Board</a></li> <li><a href="https://gshakan.wordpress.com/" title="Math Blog">George Shakan</a></li> <li><a href="http://girlsangle.wordpress.com/" title="A Math Club for Girls">Girl's Angle</a></li> <li><a href="http://godplaysdice.blogspot.com/" title="A random walk through mathematics — mostly through the random part.">God Plays Dice</a></li> <li><a href="http://scientopia.org/blogs/goodmath/" title="Finding the fun in good math; squashing bad math and the fools who promote it">Good Math, Bad Math</a></li> <li><a href="http://damekdavis.wordpress.com/" title="Damek Davis’ blog">Graduated Understanding</a></li> <li><a href="http://alanrendall.wordpress.com/" title="Alan Rendall’s blog">Hydrobates</a></li> <li><a href="https://joelmoreira.wordpress.com/" title="Joel Moreira’s math blog ">I Can't Believe It's Not Random!</a></li> <li><a href="http://yannisparissis.wordpress.com/" title="A day in the life of a mathematician. From an analysis point of view.">I Woke Up In A Strange Place</a></li> <li><a href="http://igorpak.wordpress.com/" title="Views on life and math">Igor Pak's blog</a></li> <li><a href="http://images.math.cnrs.fr/" title="La recherches mathématiques en mots et en images (in French)">Images des mathématiques</a></li> <li><a href="http://lucatrevisan.wordpress.com/" title="Luca Trevisan’s blog">In theory</a></li> <li><a href="https://www.infinitelymore.xyz/" title="The mathematics and philosophy of the infinite. By Joel David Hamkins">Infinitely more</a></li> <li><a href="http://colliand.com/blog/">James Colliander's Blog</a></li> <li><a href="http://jbuzzi.wordpress.com/">Jérôme Buzzi’s Mathematical Ramblings</a></li> <li><a href="http://jdh.hamkins.org/" title="mathematics and philosophy of the infinite">Joel David Hamkins</a></li> <li><a href="http://www.ams.org/jams/" title="A premier journal in mathematics;I am one of the editors">Journal of the American Mathematical Society</a></li> <li><a href="https://kconrad.math.uconn.edu/blurbs/">Keith Conrad's expository papers</a></li> <li><a href="http://worrydream.com/#!/KillMath" title="Mathematical visualization applets and essays by Bret Victor">Kill Math</a></li> <li><a href="http://yetaspblog.wordpress.com/" title="Yet Another Signal Processing (and Applied Math) blog">Le Petit Chercheur Illustré</a></li> <li><a href="http://lemmameringue.wordpress.com/" title="The proof is in the pudding">Lemma Meringue</a></li> <li><a href="http://lewko.wordpress.com/">Lewko's blog</a></li> <li><a href="http://blog.djalil.chafai.net/" title="WordPress weblog of Djalil Chafaï">Libres pensées d’un mathématicien ordinaire</a></li> <li><a href="http://www.lms.ac.uk/content/lms-blogs-page">LMS blogs page</a></li> <li><a href="http://ldtopology.wordpress.com/" title="Recent Progress and Open Problems">Low Dimensional Topology</a></li> <li><a href="http://m-phi.blogspot.com/" title="A blog dedicated to mathematical philosophy.">M-Phi</a></li> <li><a href="https://marksapir.wordpress.com/" title="A blog about math">Mark Sapir's blog</a></li> <li><a href="http://mathoverflow.net/">Math Overflow</a></li> <li><a href="https://www.math3ma.com/">Math3ma</a></li> <li><a href="http://mathbabe.wordpress.com" title="Exploring and venting about quantitative issues">Mathbabe</a></li> <li><a href="http://www.mathblogging.org/" title="An aggregator for maths blogs">Mathblogging</a></li> <li><a href="http://matthewkahle.wordpress.com/" title="Mathematical musings, a place for informal mathematical writings: expository, puzzles, art, and open problems. By Matthew Kahle.">Mathematical musings</a></li> <li><a href="http://www.learner.org/courses/mathilluminated/" title="A thirteen-part series for adult learners and high school teachers">Mathematics Illuminated</a></li> <li><a href="http://austmaths.wordpress.com/" title="Current issues, news, and events in Australian mathematical research and education">Mathematics in Australia</a></li> <li><a href="http://notable.math.ucdavis.edu/wiki/Mathematics_Jobs_Wiki" title="Rumours and authoritative information about the mathematics jobs market">Mathematics Jobs Wiki</a></li> <li><a href="http://math.stackexchange.com/">Mathematics Stack Exchange</a></li> <li><a href="http://micromath.wordpress.com/" title="Atomic objects, structures and concepts of mathematics">Mathematics under the Microscope</a></li> <li><a href="https://mathematicswithoutapologies.wordpress.com/" title="An unapologetic guided tour of the mathematical life, by Michael Harris">Mathematics without apologies</a></li> <li><a href="http://www.scienceblogs.de/mathlog/" title="Thilo Kuessner’s blog (in German)">Mathlog</a></li> <li><a href="http://www.mathtube.org/">Mathtube</a></li> <li><a href="http://mattbakerblog.wordpress.com/" title="Thoughts on number theory, graphs, dynamical systems, tropical geometry, pedagogy, puzzles, and the p-adics">Matt Baker's Math Blog</a></li> <li><a href="http://davidlowryduda.com/">Mixedmath</a></li> <li><a href="http://homotopical.wordpress.com/" title="Cohomology in algebraic geometry">Motivic stuff</a></li> <li><a href="http://muchado.blogli.co.il/" title="On mathematics and other animals (in Hebrew)">Much ado about nothing</a></li> <li><a href="http://scherk.pbwiki.com/" title="A wiki for online multiple choice quizzes">Multiple Choice Quiz Wiki</a></li> <li><a href="https://mycqstate.wordpress.com/" title="Thomas Vidick’s blog">MyCQstate</a></li> <li><a href="http://ncatlab.org/nlab/show/HomePage" title="Collaborative work on Mathematics, Physics and Philosophy">nLab</a></li> <li><a href="http://noncommutativegeometry.blogspot.com/" title="A group blog by several mathematicians, including Alain Connes">Noncommutative geometry blog</a></li> <li><a href="http://www.ma.utexas.edu/mediawiki/index.php/Starting_page">Nonlocal equations wiki</a></li> <li><a href="http://nuit-blanche.blogspot.com/" title="Focused on technical issues in compressed sensing and related topics">Nuit-blanche</a></li> <li><a href="http://www.numbertheory.org/" title="Resources for number theorists">Number theory web</a></li> <li><a href="https://sites.google.com/view/o-a-r-s/" title="“colloquium style” talks on topics of current interest in harmonic analysis and adjacent areas that are accessible to a general mathematical audience with basic knowledge in analysis">Online Analysis Research Seminar</a></li> <li><a href="http://outofprintmath.blogspot.com/" title="Bringing out-of-print math books into print – Vote for which book you would like to see come back into print">outofprintmath</a></li> <li><a href="https://patternsofideas.github.io/">Pattern of Ideas</a></li> <li><a href="http://pfzhang.wordpress.com/" title="Discussions related to Dynamics">Pengfei Zhang's blog</a></li> <li><a href="https://galoisrepresentations.wordpress.com/" title="Galois Representations and more!">Persiflage</a></li> <li><a href="http://cameroncounts.wordpress.com/" title="Counting the things that need to be counted">Peter Cameron's Blog</a></li> <li><a href="http://philippelefloch.wordpress.com/">Phillipe LeFloch's blog</a></li> <li><a href="http://www.proofwiki.org/wiki/Main_Page" title=" the online compendium of mathematical proofs">ProofWiki</a></li> <li><a href="http://quomodocumque.wordpress.com/" title="Jordan Ellenberg’s blog">Quomodocumque</a></li> <li><a href="https://ramismovassagh.wordpress.com/">Ramis Movassagh's blog</a></li> <li><a href="http://mathramble.wordpress.com/">Random Math</a></li> <li><a href="http://stochastix.wordpress.com/" title="scientia potentia est">Reasonable Deviations</a></li> <li><a href="http://regularize.wordpress.com/" title="Trying to keep track of what I stumble upon">Regularize</a></li> <li><a href="https://researchseminars.org/">Research Seminars</a></li> <li><a href="http://rigtriv.wordpress.com/" title="A group blog in mathematics">Rigorous Trivialities</a></li> <li><a href="http://blogs.scientificamerican.com/roots-of-unity/" title="Mathematics: learning it, doing it, celebrating it.">Roots of unity</a></li> <li><a href="https://saugatabasumath.wordpress.com/" title="Real algebraic geometry and its connections with other areas of mathematics and computer science.">Saugata Basu's blog</a></li> <li><a href="https://www.gregegan.net/SCIENCE/Science.html" title="notes on some miscellaneous scientific matters">Science Notes by Greg Egan</a></li> <li><a href="http://sbseminar.wordpress.com" title="A group maths blog at Berkeley">Secret Blogging Seminar</a></li> <li><a href="https://selectedpapers.net/">Selected Papers Network</a></li> <li><a href="http://sdenisov.wordpress.com/">Sergei Denisov's blog</a></li> <li><a href="https://dustingmixon.wordpress.com/">Short, Fat Matrices</a></li> <li><a href="http://www.scottaaronson.com/blog/" title="Scott Aaronson’s blog">Shtetl-Optimized</a></li> <li><a href="http://shaoshuanglin.wordpress.com/">Shuanglin's Blog</a></li> <li><a href="http://ifwisdomwereteachable.wordpress.com/" title="… we simply try to teach mathematics.">Since it is not…</a></li> <li><a href="http://sketchesoftopology.wordpress.com/" title="Visualizations of low dimensional topology">Sketches of topology</a></li> <li><a href="https://nttoan81.wordpress.com" title="Toan Nguyen’s blog">Snapshots in Mathematics !</a></li> <li><a href="http://kovaquestions.blogspot.com/" title="Cultural musings from a sometime mathematician.">Soft questions</a></li> <li><a href="https://backus.home.blog/" title="Aidan Backus’ blog">Some compact thoughts</a></li> <li><a href="http://math.columbia.edu/~dejong/wordpress/">Stacks Project Blog</a></li> <li><a href="http://symomega.wordpress.com/">SymOmega</a></li> <li><a href="https://blog.tanyakhovanova.com/" title="Mathematics, applications of mathematics to life in general, and my life as a mathematician.">Tanya Khovanova's Math Blog</a></li> <li><a href="http://tcsmath.wordpress.com" title="Some mathematics of theoretical computer science">tcs math</a></li> <li><a href="http://tex.stackexchange.com/" title="A MathOverflow-like site for TeX">TeX, LaTeX, and friends</a></li> <li><a href="http://ilaba.wordpress.com/" title="Izabella Laba’s blog">The accidental mathematician</a></li> <li><a href="http://thecostofknowledge.com/" title="Researchers taking a stand against Elsevier.">The Cost of Knowledge</a></li> <li><a href="http://cornellmath.wordpress.com/" title="A group maths blog at Cornell">The Everything Seminar</a></li> <li><a href="http://geomblog.blogspot.com/" title="Ruminations on computational geometry, algorithms, theoretical computer science and life">The Geomblog</a></li> <li><a href="http://www.lmfdb.org/" title="The LMFDB is an extensive database of mathematical objects arising in Number Theory.">The L-function and modular forms database</a></li> <li><a href="http://golem.ph.utexas.edu/category/" title="A group blog on math, physics and philosophy">The n-Category Café</a></li> <li><a href="http://www.noncommutative.org/" title="A noncommutative geometry group blog">The n-geometry cafe</a></li> <li><a href="http://Evan%20Chen's%20infinitely%20large%20napkin,%20aimed%20at%20making%20higher%20math%20accessible%20to%20high%20school%20students" title="https://web.evanchen.cc/napkin.html">The Napkin Project</a></li> <li><a href="http://obis.tumblr.com/">The On-Line Blog of Integer Sequences</a></li> <li><a href="http://polylogblog.wordpress.com/" title="streams, sketches, samples, sensing etc.">The polylogblog</a></li> <li><a href="http://polymathprojects.wordpress.com/" title="A central location for proposing, planning, and running polymath projects">The polymath blog</a></li> <li><a href="http://michaelnielsen.org/polymath1/index.php?title=Main_Page" title="The wiki for polymath projects.">The polymath wiki</a></li> <li><a href="http://www.tricki.org/" title="A repository of mathematical know-how">The Tricki</a></li> <li><a href="http://twofoldgaze.wordpress.com/">The twofold gaze</a></li> <li><a href="http://unapologetic.wordpress.com/" title="Mathematics for the interested.">The Unapologetic Mathematician</a></li> <li><a href="http://valuevar.wordpress.com/" title="Harald Helfgott’s blog">The value of the variable</a></li> <li><a href="https://www.mathunion.org/ceic/library/world-digital-mathematics-library-wdml" title="IMU blog">The World Digital Mathematical Library</a></li> <li><a href="http://cstheory.stackexchange.com/" title="A MathOverflow-like site for TCS">Theoretical Computer Science – StackExchange</a></li> <li><a href="https://thuses.com/" title="Math discussions">Thuses</a></li> <li><a href="http://gowers.wordpress.com" title="Mathematics related discussions (and the Princeton Companion to Mathematics)">Tim Gowers’ blog</a></li> <li><a href="http://www.dpmms.cam.ac.uk/~wtg10/mathsindex.html" title="Informal discussions of elementary (and sometimes deep) mathematics">Tim Gowers’ mathematical discussions</a></li> <li><a href="http://topologicalmusings.wordpress.com/" title="Just another mathematical blog">Todd and Vishal’s blog</a></li> <li><a href="http://vuhavan.wordpress.com/" title="In vietnamese">Van Vu's blog</a></li> <li><a href="http://vaughnclimenhaga.wordpress.com">Vaughn Climenhaga</a></li> <li><a href="http://vieuxgirondin.wordpress.com/" title="A lazy mathematician’s blog">Vieux Girondin</a></li> <li><a href="http://blogs.ams.org/visualinsight/" title="Mathematics made visible">Visual Insight</a></li> <li><a href="http://vivatsgasse7.wordpress.com/" title="A group maths blog at Bonn">Vivatsgasse 7</a></li> <li><a href="http://blogs.williams.edu/math/" title="Mathematics and Statistics at Williams College">Williams College Math/Stat Blog</a></li> <li><a href="http://windowsontheory.org" title="A research blog">Windows on Theory</a></li> <li><a href="http://www.wiskundemeisjes.nl/" title="A group maths blog from U. Leiden (in Dutch)">Wiskundemeisjes</a></li> <li><a href="http://xorshammer.wordpress.com/">XOR’s hammer</a></li> <li><a href="https://yufeizhao.wordpress.com/">Yufei Zhao's blog</a></li> <li><a href="https://extremal010101.wordpress.com/" title="Paata Ivanishvili’s blog">Zeroes and Ones</a></li> <li><a href="http://zhenghezhang.wordpress.com/">Zhenghe's Blog</a></li> </ul> </div> <div id="linkcat-203526" class="widget widget_links"><h3>Selected articles</h3> <ul class='xoxo blogroll'> <li><a href="https://terrytao.wordpress.com/2012/04/02/a-cheap-version-of-nonstandard-analysis/">A cheap version of nonstandard analysis</a></li> <li><a href="https://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/">A review of probability theory</a></li> <li><a href="https://terrytao.wordpress.com/2009/09/17/a-speech-for-the-american-academy-of-arts-and-sciences/">American Academy of Arts and Sciences speech</a></li> <li><a href="https://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/" title="A discussion of one of my favourite types of trick in analysis">Amplification, arbitrage, and the tensor power trick</a></li> <li><a href="https://terrytao.wordpress.com/2008/12/09/an-airport-inspired-puzzle/">An airport-inspired puzzle</a></li> <li><a href="https://terrytao.wordpress.com/2009/07/03/benfords-law-zipfs-law-and-the-pareto-distribution/">Benford's law, Zipf's law, and the Pareto distribution</a></li> <li><a href="https://terrytao.wordpress.com/2007/04/13/compressed-sensing-and-single-pixel-cameras/" title="A non-technical introduction to compressed sensing">Compressed sensing and single-pixel cameras</a></li> <li><a href="https://terrytao.wordpress.com/2007/12/28/einsteins-derivation-of-emc2/">Einstein’s derivation of E=mc^2</a></li> <li><a href="https://terrytao.wordpress.com/2008/12/14/on-multiple-choice-questions-in-mathematics/">On multiple choice questions in mathematics</a></li> <li><a href="https://terrytao.wordpress.com/2010/10/21/245a-problem-solving-strategies/" title="A list of problem solving strategies in analysis">Problem solving strategies</a></li> <li><a href="https://terrytao.wordpress.com/2007/02/26/quantum-mechanics-and-tomb-raider/" title="Uses Tomb Raider to construct a world with quantum-like effects.">Quantum mechanics and Tomb Raider</a></li> <li><a href="https://terrytao.wordpress.com/2010/10/21/245a-problem-solving-strategies">Real analysis problem solving strategies</a></li> <li><a href="https://terrytao.wordpress.com/2009/03/23/sailing-into-the-wind-or-faster-than-the-wind/" title="An idealised mathematical model of sailing">Sailing into the wind, or faster than the wind</a></li> <li><a href="https://terrytao.wordpress.com/2007/04/05/simons-lecture-i-structure-and-randomness-in-fourier-analysis-and-number-theory/" title="A set of three lectures on structure and randomness in mathematics">Simons lectures on structure and randomness</a></li> <li><a href="https://terrytao.wordpress.com/2008/10/10/small-samples-and-the-margin-of-error/">Small samples, and the margin of error</a></li> <li><a href="https://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/" title="A discussion of soft and hard analysis">Soft analysis, hard analysis, and the finite convergence principle</a></li> <li><a href="https://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islanders-puzzle/" title="A remarkably subtle and unintuitive logic puzzle">The blue-eyed islanders puzzle</a></li> <li><a href="https://terrytao.wordpress.com/2009/09/03/the-cosmic-distance-ladder-2/" title="Some slides on the cosmic distance ladder in astronomy">The cosmic distance ladder</a></li> <li><a href="https://terrytao.wordpress.com/2009/05/04/the-federal-budget-rescaled/">The federal budget, rescaled</a></li> <li><a href="https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/" title="Nonstandard analysis, from a hard analysis perspective.">Ultrafilters, non-standard analysis, and epsilon management</a></li> <li><a href="https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/">What is a gauge?</a></li> <li><a href="http://arxiv.org/abs/math.HO/0702396" title="An essay on what good mathematics is, and whether we should define it at all">What is good mathematics?</a></li> <li><a href="https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/" title="A detailed discussion of the global regularity problem for Navier-Stokes">Why global regularity for Navier-Stokes is hard</a></li> </ul> </div> <div id="linkcat-581" class="widget widget_links"><h3>Software</h3> <ul class='xoxo blogroll'> <li><a href="http://detexify.kirelabs.org/classify.html" title="Latex symbol classifier">Detexify</a></li> <li><a href="https://www.doi2bib.org/" title="DOI to bibtex converter">doi2bib</a></li> <li><a href="http://alexeev.org/gmailtex.html" title="An easy way to render LaTeX snippets in Gmail or Gmail chat">GmailTeX</a></li> <li><a href="http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html" title="Converts a floating point decimal into a symbolic expression">Inverse Symbolic Calculator</a></li> <li><a href="http://tams-www.informatik.uni-hamburg.de/applets/jfig/webstart.html" title="A java applet for drawing xfig-like figures">jfig</a></li> <li><a href="http://lucatrevisan.wordpress.com/latex-to-wordpress/" title="Luca Trevisan’s python script for converting LaTeX to wordpress posts">LaTeX to Wordpress</a></li> <li><a href="http://www.codecogs.com/latex/eqneditor.php">Online LaTeX Equation Editor</a></li> <li><a href="https://q.uiver.app/" title="A graphical interface for creating and editing commutative diagrams">Quiver commutative diagram editor</a></li> <li><a href="http://sagemath.blogspot.com/" title="A blog by William Stein">Sage: Open Source Mathematical Software</a></li> <li><a href="http://sbseminar.wordpress.com/2008/06/18/subverting-the-system/" title="A description of the version control software “Subversion”.">Subverting the system</a></li> </ul> </div> <div id="linkcat-702846" class="widget widget_links"><h3>The sciences</h3> <ul class='xoxo blogroll'> <li><a href="http://wiki.henryfarrell.net/wiki/index.php/Main_Page" title="A wiki for academic blogs">Academic blogs</a></li> <li><a href="http://www.amacad.org/">American Academy of Arts and Sciences</a></li> <li><a href="http://www.science.org.au/">Australian Academy of Science</a></li> <li><a href="http://blogs.discovermagazine.com/badastronomy/" title="Phil Plait’s blog">Bad Astronomy</a></li> <li><a href="http://www.nasonline.org/site/PageServer">National Academy of Science</a></li> <li><a href="http://www.realclimate.org/" title="Climate science, minus (most of the) politics">RealClimate</a></li> <li><a href="http://www.schneier.com/blog/" title="Bruce Schneier on security issues">Schneier on security</a></li> <li><a href="http://www.sciencebasedmedicine.org/" title="Exploring issues and controversies in the relationship between science and medicine">Science-Based Medicine</a></li> <li><a href="http://chronicle.com/free/v49/i21/21b02001.htm" title="A Chronicle article from 2003; still timely">Seven warning signs of bogus science</a></li> <li><a href="http://royalsociety.org/">The Royal Society</a></li> <li><a href="http://blog.lib.umn.edu/denis036/thisweekinevolution/" title="Recent papers in evolutionary biology">This week in evolution</a></li> <li><a href="http://www.tolweb.org/tree/" title="Explore the tree of life">Tree of Life Web Project</a></li> </ul> </div> <div id="top-posts-2" class="widget widget_top-posts"><h3>Top Posts</h3><ul><li><a href="https://terrytao.wordpress.com/2025/02/13/cosmic-distance-ladder-video-with-grant-sanderson-3blue1brown-commentary-and-corrections/" class="bump-view" data-bump-view="tp">Cosmic Distance Ladder video with Grant Sanderson (3blue1brown): commentary and corrections</a></li><li><a href="https://terrytao.wordpress.com/career-advice/" class="bump-view" data-bump-view="tp">Career advice</a></li><li><a href="https://terrytao.wordpress.com/2025/01/28/new-exponent-pairs-zero-density-estimates-and-zero-additive-energy-estimates-a-systematic-approach/" class="bump-view" data-bump-view="tp">New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach</a></li><li><a href="https://terrytao.wordpress.com/books/climbing-the-cosmic-distance-ladder/" class="bump-view" data-bump-view="tp">Climbing the cosmic distance ladder</a></li><li><a href="https://terrytao.wordpress.com/books/" class="bump-view" data-bump-view="tp">Books</a></li><li><a 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href="https://terrytao.wordpress.com/tag/universality/" class="tag-cloud-link tag-link-553131 tag-link-position-72" style="font-size: 15.21875pt;" aria-label="universality (16 items)">universality</a> <a href="https://terrytao.wordpress.com/tag/van-vu/" class="tag-cloud-link tag-link-9482008 tag-link-position-73" style="font-size: 17.1875pt;" aria-label="Van Vu (20 items)">Van Vu</a> <a href="https://terrytao.wordpress.com/tag/wave-maps/" class="tag-cloud-link tag-link-1830969 tag-link-position-74" style="font-size: 8pt;" aria-label="wave maps (7 items)">wave maps</a> <a href="https://terrytao.wordpress.com/tag/yitang-zhang/" class="tag-cloud-link tag-link-168303021 tag-link-position-75" style="font-size: 9.09375pt;" aria-label="Yitang Zhang (8 items)">Yitang Zhang</a></div><div id="rss-2" class="widget widget_rss"><h3><a class="rsswidget" href="https://polymathprojects.org/feed/" title="Syndicate this content"><img style="background: orange; color: white; border: none;" width="14" height="14" src="https://s.wordpress.com/wp-includes/images/rss.png?m=1354137473i" alt="RSS" /></a> <a class="rsswidget" href="https://polymathprojects.org" title="Massively collaborative mathematical projects">The Polymath Blog</a></h3><ul><li><a class='rsswidget' href='https://polymathprojects.org/2021/02/20/polymath-projects-2021/' title=' After the success of Polymath1 and the launching of Polymath3 and Polymath4, Tim Gowers wrote a blog post “Possible future Polymath projects” for planning the next polymath project on his blog. The post mentioned 9 possible projects. (Four of them later turned to polymath projects.) Following the post and separate posts describing some of […]'>Polymath projects 2021</a></li><li><a class='rsswidget' href='https://polymathprojects.org/2019/06/09/a-sort-of-polymath-on-a-famous-mathoverflow-problem/' title=' Is there any polynomials of two variables with rational coefficients, such that the map is a bijection? This is a famous 9-years old open question on MathOverflow. Terry Tao initiated a sort of polymath attempt to solve this problem conditioned on some conjectures from arithmetic algebraic geometry. This project is based on an plan […]'>A sort of Polymath on a famous MathOverflow problem</a></li><li><a class='rsswidget' href='https://polymathprojects.org/2019/02/03/ten-years-of-polymath/' title='Ten years ago on January 27, 2009, Polymath1 was proposed by Tim Gowers and was launched on February 1, 2009. The first project was successful and it followed by 15 other formal polymath projects and a few other projects of similar nature.'>Ten Years of Polymath</a></li><li><a class='rsswidget' href='https://polymathprojects.org/2018/10/19/updates-and-pictures/' title='Three short items: Progress on Rota’s conjecture (polymath12) by Bucić, Kwan, Pokrovskiy, and Sudakov First, there is a remarkable development on Rota’s basis conjecture (Polymath12) described in the paper Halfway to Rota’s basis conjecture, by Matija Bucić, Matthew Kwan, Alexey Pokrovskiy, and Benny Sudakov Abstract: In 1989, Rota made the following conject […]'>Updates and Pictures</a></li><li><a class='rsswidget' href='https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/' title='The Hadwiger-Nelson problem is that of determining the chromatic number of the plane (), defined as the minimum number of colours that can be assigned to the points of the plane so as to prevent any two points unit distance apart from being the same colour. It was first posed in 1950 and the bounds […]'>Polymath proposal: finding simpler unit distance graphs of chromatic number 5</a></li><li><a class='rsswidget' href='https://polymathprojects.org/2018/01/26/a-new-polymath-proposal-related-to-the-riemann-hypothesis-over-taos-blog/' title='(From a post “the music of the primes” by Marcus du Sautoy.) A new polymath proposal over Terry Tao’s blog who wrote: “Building on the interest expressed in the comments to this previous post, I am now formally proposing to initiate a “Polymath project” on the topic of obtaining new upper bounds on the de Bruijn-Newman constant . […]'>A new polymath proposal (related to the Riemann Hypothesis) over Tao’s blog</a></li><li><a class='rsswidget' href='https://polymathprojects.org/2018/01/26/spontaneous-polymath-14-a-success/' title='This post is to report an unplanned polymath project, now called polymath 14 that took place over Terry Tao’s blog. A problem was posed by Apoorva Khare was presented and discussed and openly and collectively solved. (And the paper arxived.)'>Spontaneous Polymath 14 – A success!</a></li><li><a class='rsswidget' href='https://polymathprojects.org/2017/08/22/polymath-13-a-success/' title='This post is to note that the polymath13 project has successfully settled one of the major objective. Reports on it can be found on Gower’s blog especially in this post Intransitive dice IV: first problem more or less solved? and this post Intransitive dice VI: sketch proof of the main conjecture for the balanced-sequences model. '>Polymath 13 – a success!</a></li><li><a class='rsswidget' href='https://polymathprojects.org/2017/05/15/non-transitive-dice-over-gowerss-blog/' title='A polymath-style project on non transitive dice (Wikipedea) is now running over Gowers blog. (Here is the link to the first post.) '>Non-transitive Dice over Gowers’s Blog</a></li><li><a class='rsswidget' href='https://polymathprojects.org/2017/05/05/rotas-basis-conjecture-polymath-12-post-3/' title='We haven’t quite hit the 100-comment mark on the second Polymath 12 blog post, but this seems like a good moment to take stock. The project has lost some of its initial momentum, perhaps because other priorities have intruded into the lives of the main participants (I know that this is true of myself). However, […]'>Rota’s Basis Conjecture: Polymath 12, post 3</a></li></ul></div> </div> </div> <div id="comments" class="commentlist"> <div id="comments-meta"> <h2 class="comments-title">221 comments</h2> <p class="comments-feed"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/feed/">Comments feed for this article</a></p> </div> <div class="pingback even thread-even depth-1 vcard" id="comment-477265"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-477265" title="Permalink to this comment">27 January, 2017 at 12:08 pm</a></p> <p class="comment-author"><strong class="fn"><a href="https://unavistacircular.wordpress.com/2017/01/27/que-sorpresas-esconden-las-sumas-infinitas-i/" class="url" rel="ugc external nofollow">¿Qué sorpresas esconden las sumas infinitas? I | Una vista circular</a></strong></p> </div> <div class="comment-content"> <p>[…] 1 + 2 + 3 + 4 + ….. Yo he aprendido este enfoque, que quizás no es aún muy conocido, en un magnífico post en el blog de Terence Tao. Mi objetivo en esta serie de posts es dar los detalles imprescindibles […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_477265"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=477265#respond" data-commentid="477265" data-postid="3643" data-belowelement="comment-477265" data-respondelement="respond" data-replyto="Reply to ¿Qué sorpresas esconden las sumas infinitas? I | Una vista circular" aria-label="Reply to ¿Qué sorpresas esconden las sumas infinitas? I | Una vista circular">Reply</a> </div> </div> <div class="comment byuser comment-author-kolmogorovscale odd alt thread-odd thread-alt depth-1 vcard" id="comment-477319"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-477319" title="Permalink to this comment">30 January, 2017 at 9:25 pm</a></p> <p class="comment-author"><strong class="fn"><a href="http://darkgreymatterthings.wordpress.com" class="url" rel="ugc external nofollow">kolmogorovscale</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://2.gravatar.com/avatar/29bbfce11b8c6f648341eaf99d5c68b7dbe47ad0de99fb2b4ee6606ef1e72d33?s=32&d=identicon&r=PG' srcset='https://2.gravatar.com/avatar/29bbfce11b8c6f648341eaf99d5c68b7dbe47ad0de99fb2b4ee6606ef1e72d33?s=32&d=identicon&r=PG 1x, https://2.gravatar.com/avatar/29bbfce11b8c6f648341eaf99d5c68b7dbe47ad0de99fb2b4ee6606ef1e72d33?s=48&d=identicon&r=PG 1.5x, https://2.gravatar.com/avatar/29bbfce11b8c6f648341eaf99d5c68b7dbe47ad0de99fb2b4ee6606ef1e72d33?s=64&d=identicon&r=PG 2x, https://2.gravatar.com/avatar/29bbfce11b8c6f648341eaf99d5c68b7dbe47ad0de99fb2b4ee6606ef1e72d33?s=96&d=identicon&r=PG 3x, https://2.gravatar.com/avatar/29bbfce11b8c6f648341eaf99d5c68b7dbe47ad0de99fb2b4ee6606ef1e72d33?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Reblogged this on <a href="https://statphysbiochem.wordpress.com/2017/01/31/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/" rel="nofollow">Stat Phys Bio Chem</a>.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_477319"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=477319#respond" data-commentid="477319" data-postid="3643" data-belowelement="comment-477319" data-respondelement="respond" data-replyto="Reply to kolmogorovscale" aria-label="Reply to kolmogorovscale">Reply</a> </div> </div> <div class="pingback even thread-even depth-1 vcard" id="comment-483139"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-483139" title="Permalink to this comment">25 June, 2017 at 3:51 pm</a></p> <p class="comment-author"><strong class="fn"><a href="http://minusoneby12.com/2017/06/18/__trashed/" class="url" rel="ugc external nofollow">Why -1/12? – Maths and Machine Learning</a></strong></p> </div> <div class="comment-content"> <p>[…] very thorough (and reasonably accessible) discussion of the above series can be found in Terrence Tao’s blog post. In this post, I will not be attempting to either match the rigor or the comprehensiveness of […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_483139"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=483139#respond" data-commentid="483139" data-postid="3643" data-belowelement="comment-483139" data-respondelement="respond" data-replyto="Reply to Why -1/12? – Maths and Machine Learning" aria-label="Reply to Why -1/12? – Maths and Machine Learning">Reply</a> </div> </div> <div class="comment odd alt thread-odd thread-alt depth-1 vcard" id="comment-484116"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-484116" title="Permalink to this comment">22 July, 2017 at 7:03 am</a></p> <p class="comment-author"><strong class="fn">Dr. Enoch Opeyemi</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/063fc5836dd1cff42cc76257e9647f5fc4784b6a6d9f375760446f81f950483b?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/063fc5836dd1cff42cc76257e9647f5fc4784b6a6d9f375760446f81f950483b?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/063fc5836dd1cff42cc76257e9647f5fc4784b6a6d9f375760446f81f950483b?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/063fc5836dd1cff42cc76257e9647f5fc4784b6a6d9f375760446f81f950483b?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/063fc5836dd1cff42cc76257e9647f5fc4784b6a6d9f375760446f81f950483b?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/063fc5836dd1cff42cc76257e9647f5fc4784b6a6d9f375760446f81f950483b?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Greetings Prof. Tao,<br /> Over the years,I have been working on the Riemann Hypothesis and I have recently found some results that might be interesting to you sir.</p> <p>I would like your comments and contributions on them because I got to a point at which I need to use analytic continuation method on my finding.</p> <p>Thank you Sir!</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_484116"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=484116#respond" data-commentid="484116" data-postid="3643" data-belowelement="comment-484116" data-respondelement="respond" data-replyto="Reply to Dr. Enoch Opeyemi" aria-label="Reply to Dr. Enoch Opeyemi">Reply</a> </div> </div> <div class="pingback even thread-even depth-1 vcard" id="comment-485932"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-485932" title="Permalink to this comment">6 September, 2017 at 6:04 am</a></p> <p class="comment-author"><strong class="fn"><a href="https://thiagrr.wordpress.com/2017/09/06/zeta-function-regularization/" class="url" rel="ugc external nofollow">Zeta function regularization – Amplitudes. . .</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://thiagrr.files.wordpress.com/2017/10/cropped-cropped-14445056_1791238121124090_2177132845965308650_o.jpg?w=32' srcset='https://thiagrr.files.wordpress.com/2017/10/cropped-cropped-14445056_1791238121124090_2177132845965308650_o.jpg?w=32 1x, https://thiagrr.files.wordpress.com/2017/10/cropped-cropped-14445056_1791238121124090_2177132845965308650_o.jpg?w=48 1.5x, https://thiagrr.files.wordpress.com/2017/10/cropped-cropped-14445056_1791238121124090_2177132845965308650_o.jpg?w=64 2x, https://thiagrr.files.wordpress.com/2017/10/cropped-cropped-14445056_1791238121124090_2177132845965308650_o.jpg?w=96 3x, https://thiagrr.files.wordpress.com/2017/10/cropped-cropped-14445056_1791238121124090_2177132845965308650_o.jpg?w=128 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>[…] you can read about divergent series in the following post by Terence Tao and in the book Divergent Series by Hardy. The Riemann hypothesis is an open problem in math, and […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_485932"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=485932#respond" data-commentid="485932" data-postid="3643" data-belowelement="comment-485932" data-respondelement="respond" data-replyto="Reply to Zeta function regularization – Amplitudes. . ." aria-label="Reply to Zeta function regularization – Amplitudes. . .">Reply</a> </div> </div> <div class="comment odd alt thread-odd thread-alt depth-1 vcard" id="comment-486030"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-486030" title="Permalink to this comment">9 September, 2017 at 2:00 am</a></p> <p class="comment-author"><strong class="fn"><a href="https://www.facebook.com/app_scoped_user_id/1469358700/" class="url" rel="ugc external nofollow">Jose Javier Garcia Moreta</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://graph.facebook.com/v2.9/1469358700/picture?type=large&_md5=337506ed9fd7b2dd165ab3accc376073' srcset='https://graph.facebook.com/v2.9/1469358700/picture?type=large&_md5=337506ed9fd7b2dd165ab3accc376073 1x, https://graph.facebook.com/v2.9/1469358700/picture?type=large&_md5=337506ed9fd7b2dd165ab3accc376073 1.5x, https://graph.facebook.com/v2.9/1469358700/picture?type=large&_md5=337506ed9fd7b2dd165ab3accc376073 2x, https://graph.facebook.com/v2.9/1469358700/picture?type=large&_md5=337506ed9fd7b2dd165ab3accc376073 3x, https://graph.facebook.com/v2.9/1469358700/picture?type=large&_md5=337506ed9fd7b2dd165ab3accc376073 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Dear professor tao.. for the case of the general harmonic series</p> <p>\sum_{n=0}^{\infty} \frac{1}{(n+a)}? is this \-Psi(a) or \|-\Psi (a)+log(a) ?? </p> <p>which one is the correct regularized result ??</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_486030"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=486030#respond" data-commentid="486030" data-postid="3643" data-belowelement="comment-486030" data-respondelement="respond" data-replyto="Reply to Jose Javier Garcia Moreta" aria-label="Reply to Jose Javier Garcia Moreta">Reply</a> </div> </div> <div class="comment even thread-even depth-1 vcard" id="comment-486340"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-486340" title="Permalink to this comment">17 September, 2017 at 11:52 am</a></p> <p class="comment-author"><strong class="fn">Bernard Montaron</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/09c2e82b9d56b1d2d001dd88abd423bac8be566f07e47463591adbadf731c3d4?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/09c2e82b9d56b1d2d001dd88abd423bac8be566f07e47463591adbadf731c3d4?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/09c2e82b9d56b1d2d001dd88abd423bac8be566f07e47463591adbadf731c3d4?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/09c2e82b9d56b1d2d001dd88abd423bac8be566f07e47463591adbadf731c3d4?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/09c2e82b9d56b1d2d001dd88abd423bac8be566f07e47463591adbadf731c3d4?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/09c2e82b9d56b1d2d001dd88abd423bac8be566f07e47463591adbadf731c3d4?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Here is another strange equality, which might be compared to equation (7):<br /> <img src="https://s0.wp.com/latex.php?latex=S%3D%5Csum_%7Bk%3D0%7D%5E%7B%5Cinfty+%7D10%5Ek%3D-%5Cfrac%7B1%7D%7B9%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=S%3D%5Csum_%7Bk%3D0%7D%5E%7B%5Cinfty+%7D10%5Ek%3D-%5Cfrac%7B1%7D%7B9%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=S%3D%5Csum_%7Bk%3D0%7D%5E%7B%5Cinfty+%7D10%5Ek%3D-%5Cfrac%7B1%7D%7B9%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="S=\sum_{k=0}^{\infty }10^k=-\frac{1}{9}" class="latex" /><br /> And here is the proof of this: Clearly 10S = S – 1 (!)<br /> And since 1/9 = 0.111111… this leads to<br /> <img src="https://s0.wp.com/latex.php?latex=S%3D%5Csum_%7B-%5Cinfty%7D%5E%7B%5Cinfty+%7D10%5Ek+%3D+0&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=S%3D%5Csum_%7B-%5Cinfty%7D%5E%7B%5Cinfty+%7D10%5Ek+%3D+0&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=S%3D%5Csum_%7B-%5Cinfty%7D%5E%7B%5Cinfty+%7D10%5Ek+%3D+0&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="S=\sum_{-\infty}^{\infty }10^k = 0" class="latex" /><br /> You can generalize this to any rational number, like e.g.<br /> <img src="https://s0.wp.com/latex.php?latex=142857+%5Csum_%7Bk%3D0%7D%5E%7B%5Cinfty+%7D10%5E%7B6k%7D%3D-%5Cfrac%7B1%7D%7B7%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=142857+%5Csum_%7Bk%3D0%7D%5E%7B%5Cinfty+%7D10%5E%7B6k%7D%3D-%5Cfrac%7B1%7D%7B7%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=142857+%5Csum_%7Bk%3D0%7D%5E%7B%5Cinfty+%7D10%5E%7B6k%7D%3D-%5Cfrac%7B1%7D%7B7%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="142857 \sum_{k=0}^{\infty }10^{6k}=-\frac{1}{7}" class="latex" /><br /> And with 1/7 = 0.142857142857… it follows that<br /> <img src="https://s0.wp.com/latex.php?latex=142857+%5Csum_%7B-%5Cinfty%7D%5E%7B%5Cinfty+%7D10%5E%7B6k%7D+%3D+0&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=142857+%5Csum_%7B-%5Cinfty%7D%5E%7B%5Cinfty+%7D10%5E%7B6k%7D+%3D+0&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=142857+%5Csum_%7B-%5Cinfty%7D%5E%7B%5Cinfty+%7D10%5E%7B6k%7D+%3D+0&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="142857 \sum_{-\infty}^{\infty }10^{6k} = 0" class="latex" /><br /> Sorry about this!</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_486340"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=486340#respond" data-commentid="486340" data-postid="3643" data-belowelement="comment-486340" data-respondelement="respond" data-replyto="Reply to Bernard Montaron" aria-label="Reply to Bernard Montaron">Reply</a> </div> </div> <div class="comment odd alt thread-odd thread-alt depth-1 vcard parent" id="comment-488304"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-488304" title="Permalink to this comment">1 November, 2017 at 9:13 am</a></p> <p class="comment-author"><strong class="fn">B. Koller</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/921acfc39458891bf79a4af7b8bf02e4d30b2d4408dad8f23aaf088b4509fbb3?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/921acfc39458891bf79a4af7b8bf02e4d30b2d4408dad8f23aaf088b4509fbb3?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/921acfc39458891bf79a4af7b8bf02e4d30b2d4408dad8f23aaf088b4509fbb3?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/921acfc39458891bf79a4af7b8bf02e4d30b2d4408dad8f23aaf088b4509fbb3?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/921acfc39458891bf79a4af7b8bf02e4d30b2d4408dad8f23aaf088b4509fbb3?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/921acfc39458891bf79a4af7b8bf02e4d30b2d4408dad8f23aaf088b4509fbb3?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>A question of terminology :</p> <p>I have a rather naive question. We can prove in a strictly mathematical sense that the sum over all the natural numbers must be positive, for instance by induction. If on the other side we put the sum of all natural numbers equal to -1/12 then we prove the contradiction -1/12 > 0 and one knows that from a contradiction everything follows.<br /> So should we not write instead of the equal sign =, which stands for an equivalence relation and is therefore transitive, the sign : from logic which is in my understanding not an equivalence relation and is not giving a contradiction. I know this is not done usually in mathematics, but it would have the advantage to create less misunderstandings. </p> <p>Thanks</p> <p>PS. Prof. Tao approach gave me a much better understanding of infinite series, thanks a lot.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_488304"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=488304#respond" data-commentid="488304" data-postid="3643" data-belowelement="comment-488304" data-respondelement="respond" data-replyto="Reply to B. Koller" aria-label="Reply to B. Koller">Reply</a> </div> </div> <ul class="children"> <div class="comment even depth-2 vcard parent" id="comment-489087"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-489087" title="Permalink to this comment">21 November, 2017 at 8:42 pm</a></p> <p class="comment-author"><strong class="fn">Ángel Méndez Rivera</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>“We can prove in a strictly mathematical sense that the sum over all the natural numbers must be positive, for instance by induction.”</p> <p>This is not true. You can only prove by induction that the sum of finitely many addends, all of which are elements of set of natural numbers, must itself be a natural number. However, this proof cannot be executed with infinitely many addends, since it can be rigorously proven that infinite sums are in general not commutative, nor are they closed under any subset of the real numbers, nor are they associative. </p> <p>“If on the other side we put the sum of all natural numbers equal to -1/12 then we prove the contradiction -1/12 > 0”</p> <p>You’re assuming the sum is greater than zero, which is not. Terence Tao provided an explanation of why this is the case. Whenever N is finite, the term of the highest order in asymptotic expansion, C(η,1)N^2, is always positive and larger in magnitude than 1/12. However, this can be shown false whenever N is infinite. </p> <p>“So should we not write instead of the equal sign =, which stands for an equivalence relation and is therefore transitive, the sign : from logic which is in my understanding not an equivalence relation and is not giving a contradiction.”</p> <p>There is no contradiction. Mathematicians have been using = for centuries, and the reason is that the usage of it is not wrong. </p> <p>“I know this is not done usually in mathematics, but it would have the advantage to create less misunderstandings.”</p> <p>There is no misunderstanding. The = is meant literally here, because there are methods to prove that both expressions in the equation are EQUAL, they do not merely correspond to each other, but are actually equivalent.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_489087"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=489087#respond" data-commentid="489087" data-postid="3643" data-belowelement="comment-489087" data-respondelement="respond" data-replyto="Reply to Ángel Méndez Rivera" aria-label="Reply to Ángel Méndez Rivera">Reply</a> </div> </div> <ul class="children"> <div class="comment odd alt depth-3 vcard" id="comment-679757"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-679757" title="Permalink to this comment">15 July, 2023 at 1:38 pm</a></p> <p class="comment-author"><strong class="fn"><a href="https://www.hs-augsburg.de/~mueckenh/" class="url" rel="ugc external nofollow">Wolfgang Mückenheim</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/fcfce52d7b52f746cc1a25e97ac9a35d3be0f10b1da7e79c78fd8ac0311e2793?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/fcfce52d7b52f746cc1a25e97ac9a35d3be0f10b1da7e79c78fd8ac0311e2793?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/fcfce52d7b52f746cc1a25e97ac9a35d3be0f10b1da7e79c78fd8ac0311e2793?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/fcfce52d7b52f746cc1a25e97ac9a35d3be0f10b1da7e79c78fd8ac0311e2793?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/fcfce52d7b52f746cc1a25e97ac9a35d3be0f10b1da7e79c78fd8ac0311e2793?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/fcfce52d7b52f746cc1a25e97ac9a35d3be0f10b1da7e79c78fd8ac0311e2793?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Adding positive values finitely or infinitely often will never result in a decrease.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_679757"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=679757#respond" data-commentid="679757" data-postid="3643" data-belowelement="comment-679757" data-respondelement="respond" data-replyto="Reply to Wolfgang Mückenheim" aria-label="Reply to Wolfgang Mückenheim">Reply</a> </div> </div> </ul><!-- .children --> <div class="comment even depth-2 vcard" id="comment-505752"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-505752" title="Permalink to this comment">8 October, 2018 at 2:21 am</a></p> <p class="comment-author"><strong class="fn">Ángel Méndez Rivera</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Also, to add to my response to the comment, your claim that everything follows from a contradiction is not true. The principle of explosion is merely an axiom of classical propositional logic, but arithmetic first-order logic uses a Hilbert-style deduction system, in which the principle of explosion cannot be proven nor disproven.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_505752"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=505752#respond" data-commentid="505752" data-postid="3643" data-belowelement="comment-505752" data-respondelement="respond" data-replyto="Reply to Ángel Méndez Rivera" aria-label="Reply to Ángel Méndez Rivera">Reply</a> </div> </div> </ul><!-- .children --> <div class="pingback odd alt thread-even depth-1 vcard" id="comment-489375"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-489375" title="Permalink to this comment">30 November, 2017 at 3:17 pm</a></p> <p class="comment-author"><strong class="fn"><a href="https://mathswithapinchofsalt.wordpress.com/2017/11/30/1234-1-12/" class="url" rel="ugc external nofollow">1+2+3+4+ … = –1/12 | Maths with a Pinch of Salt</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://mathswithapinchofsalt.files.wordpress.com/2017/02/cropped-nasa-salt-crystals.jpg?w=32' srcset='https://mathswithapinchofsalt.files.wordpress.com/2017/02/cropped-nasa-salt-crystals.jpg?w=32 1x, https://mathswithapinchofsalt.files.wordpress.com/2017/02/cropped-nasa-salt-crystals.jpg?w=48 1.5x, https://mathswithapinchofsalt.files.wordpress.com/2017/02/cropped-nasa-salt-crystals.jpg?w=64 2x, https://mathswithapinchofsalt.files.wordpress.com/2017/02/cropped-nasa-salt-crystals.jpg?w=96 3x, https://mathswithapinchofsalt.files.wordpress.com/2017/02/cropped-nasa-salt-crystals.jpg?w=128 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>[…] Terence Tao (o różnych sposobach „sumowania” nieskończonych sum) […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_489375"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=489375#respond" data-commentid="489375" data-postid="3643" data-belowelement="comment-489375" data-respondelement="respond" data-replyto="Reply to 1+2+3+4+ … = –1/12 | Maths with a Pinch of Salt" aria-label="Reply to 1+2+3+4+ … = –1/12 | Maths with a Pinch of Salt">Reply</a> </div> </div> <div class="pingback even thread-odd thread-alt depth-1 vcard" id="comment-489891"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-489891" title="Permalink to this comment">14 December, 2017 at 1:26 pm</a></p> <p class="comment-author"><strong class="fn"><a href="https://mathematicacumgranosalis.wordpress.com/2017/12/14/1234-1-12/" class="url" rel="ugc external nofollow">1+2+3+4+ … = –1/12 – Mathematica cum grano salis</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://mathematicacumgranosalis.files.wordpress.com/2017/12/cropped-208_wurgap1.jpg?w=32' srcset='https://mathematicacumgranosalis.files.wordpress.com/2017/12/cropped-208_wurgap1.jpg?w=32 1x, https://mathematicacumgranosalis.files.wordpress.com/2017/12/cropped-208_wurgap1.jpg?w=48 1.5x, https://mathematicacumgranosalis.files.wordpress.com/2017/12/cropped-208_wurgap1.jpg?w=64 2x, https://mathematicacumgranosalis.files.wordpress.com/2017/12/cropped-208_wurgap1.jpg?w=96 3x, https://mathematicacumgranosalis.files.wordpress.com/2017/12/cropped-208_wurgap1.jpg?w=128 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>[…] Terence Tao (on the many ways of making sense of „impossible” sums) […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_489891"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=489891#respond" data-commentid="489891" data-postid="3643" data-belowelement="comment-489891" data-respondelement="respond" data-replyto="Reply to 1+2+3+4+ … = –1/12 – Mathematica cum grano salis" aria-label="Reply to 1+2+3+4+ … = –1/12 – Mathematica cum grano salis">Reply</a> </div> </div> <div class="comment byuser comment-author-ghoshadi odd alt thread-even depth-1 vcard" id="comment-493438"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-493438" title="Permalink to this comment">3 March, 2018 at 5:27 am</a></p> <p class="comment-author"><strong class="fn"><a href="http://ghoshadi.wordpress.com" class="url" rel="ugc external nofollow">Aditya Ghosh</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://1.gravatar.com/avatar/1a821b07175b009f2420f608551dc957bcfb350102af3baedb585137ce5c12ee?s=32&d=identicon&r=PG' srcset='https://1.gravatar.com/avatar/1a821b07175b009f2420f608551dc957bcfb350102af3baedb585137ce5c12ee?s=32&d=identicon&r=PG 1x, https://1.gravatar.com/avatar/1a821b07175b009f2420f608551dc957bcfb350102af3baedb585137ce5c12ee?s=48&d=identicon&r=PG 1.5x, https://1.gravatar.com/avatar/1a821b07175b009f2420f608551dc957bcfb350102af3baedb585137ce5c12ee?s=64&d=identicon&r=PG 2x, https://1.gravatar.com/avatar/1a821b07175b009f2420f608551dc957bcfb350102af3baedb585137ce5c12ee?s=96&d=identicon&r=PG 3x, https://1.gravatar.com/avatar/1a821b07175b009f2420f608551dc957bcfb350102af3baedb585137ce5c12ee?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Reblogged this on <a href="https://mathsupportweb.wordpress.com/2018/03/03/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/" rel="nofollow">Mathematics Support</a>.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_493438"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=493438#respond" data-commentid="493438" data-postid="3643" data-belowelement="comment-493438" data-respondelement="respond" data-replyto="Reply to Aditya Ghosh" aria-label="Reply to Aditya Ghosh">Reply</a> </div> </div> <div class="comment even thread-odd thread-alt depth-1 vcard" id="comment-497671"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-497671" title="Permalink to this comment">3 May, 2018 at 1:44 am</a></p> <p class="comment-author"><strong class="fn">Clément Caubel</strong></p> </div> <div class="comment-content"> <img alt='' src='https://2.gravatar.com/avatar/8bc7c6e93a8a2bcec90075c95f4ed41f6fd275095fe5ecba82d671845a4d2e26?s=32&d=identicon&r=PG' srcset='https://2.gravatar.com/avatar/8bc7c6e93a8a2bcec90075c95f4ed41f6fd275095fe5ecba82d671845a4d2e26?s=32&d=identicon&r=PG 1x, https://2.gravatar.com/avatar/8bc7c6e93a8a2bcec90075c95f4ed41f6fd275095fe5ecba82d671845a4d2e26?s=48&d=identicon&r=PG 1.5x, https://2.gravatar.com/avatar/8bc7c6e93a8a2bcec90075c95f4ed41f6fd275095fe5ecba82d671845a4d2e26?s=64&d=identicon&r=PG 2x, https://2.gravatar.com/avatar/8bc7c6e93a8a2bcec90075c95f4ed41f6fd275095fe5ecba82d671845a4d2e26?s=96&d=identicon&r=PG 3x, https://2.gravatar.com/avatar/8bc7c6e93a8a2bcec90075c95f4ed41f6fd275095fe5ecba82d671845a4d2e26?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Just to mention a little mistake/typo: after the proof of the Euler-MacLaurin formula (22), when deducing the claim (14), the left hand side is <img src="https://s0.wp.com/latex.php?latex=c_%7B%5Ceta%2Cs%7DN%5E%7Bs%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=c_%7B%5Ceta%2Cs%7DN%5E%7Bs%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=c_%7B%5Ceta%2Cs%7DN%5E%7Bs%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="c_{\eta,s}N^{s+1}" class="latex" /> (and not <img src="https://s0.wp.com/latex.php?latex=c_%7B%5Ceta%2Cs%7DN%5Es&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=c_%7B%5Ceta%2Cs%7DN%5Es&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=c_%7B%5Ceta%2Cs%7DN%5Es&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="c_{\eta,s}N^s" class="latex" />). Also, the size <img src="https://s0.wp.com/latex.php?latex=C_%7B%5Ceta%2Cs%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=C_%7B%5Ceta%2Cs%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=C_%7B%5Ceta%2Cs%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="C_{\eta,s}" class="latex" /> is used during the introduction, and then replaced by <img src="https://s0.wp.com/latex.php?latex=c_%7B%5Ceta%2Cs%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=c_%7B%5Ceta%2Cs%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=c_%7B%5Ceta%2Cs%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="c_{\eta,s}" class="latex" />.</p> <p>This gives me the opportunity to thank you warmly for sharing your notes on this blog!</p> <p><i>[Corrected, thanks -T.]</i></p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_497671"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=497671#respond" data-commentid="497671" data-postid="3643" data-belowelement="comment-497671" data-respondelement="respond" data-replyto="Reply to Clément Caubel" aria-label="Reply to Clément Caubel">Reply</a> </div> </div> <div class="pingback odd alt thread-even depth-1 vcard" id="comment-502442"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-502442" title="Permalink to this comment">7 July, 2018 at 1:32 pm</a></p> <p class="comment-author"><strong class="fn"><a href="https://www.physicsforums.com/threads/is-string-theory-built-on-funny-math.951077/#post-6023261" class="url" rel="ugc external nofollow">Is String Theory Built On Funny Math? | Physics Forums</a></strong></p> </div> <div class="comment-content"> <p>[…] if you want more heavy math see the blog by Terry Tao the guy that started me thinking about this: <a href="https://terrytao.wordpress.com/2010" rel="ugc">https://terrytao.wordpress.com/2010</a>…tion-and-real-variable-analytic-continuation/ Now the 64 million dollar question is this, Look in video 1 – he opens a string theory text – low […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_502442"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=502442#respond" data-commentid="502442" data-postid="3643" data-belowelement="comment-502442" data-respondelement="respond" data-replyto="Reply to Is String Theory Built On Funny Math? | Physics Forums" aria-label="Reply to Is String Theory Built On Funny Math? | Physics Forums">Reply</a> </div> </div> <div class="pingback even thread-odd thread-alt depth-1 vcard" id="comment-503565"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-503565" title="Permalink to this comment">1 August, 2018 at 8:00 am</a></p> <p class="comment-author"><strong class="fn"><a href="https://thisweekifoundout.wordpress.com/2018/08/01/the-sum-of-all-numbers-is-1-12/" class="url" rel="ugc external nofollow">The Sum Of All Numbers Is -1/12? – This Week I Found Out</a></strong></p> </div> <div class="comment-content"> <p>[…] <a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-fun&#8230" rel="ugc">https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-fun&#8230</a>; […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_503565"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=503565#respond" data-commentid="503565" data-postid="3643" data-belowelement="comment-503565" data-respondelement="respond" data-replyto="Reply to The Sum Of All Numbers Is -1/12? – This Week I Found Out" aria-label="Reply to The Sum Of All Numbers Is -1/12? – This Week I Found Out">Reply</a> </div> </div> <div class="pingback odd alt thread-even depth-1 vcard" id="comment-503674"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-503674" title="Permalink to this comment">5 August, 2018 at 9:41 am</a></p> <p class="comment-author"><strong class="fn"><a href="http://maddmaths.simai.eu/rubriche/fake_papers/fake-papers1_riemann/" class="url" rel="ugc external nofollow">Fake papers #1: L’uomo che scambiò la Zeta di Riemann per un polinomio | Maddmaths!</a></strong></p> </div> <div class="comment-content"> <p>[…] Per farsi un’idea di come questa procedura sia di difficile comprensione per i non esperti, usando la continuazione analitica si arriva a scrivere [1+2+3+4+dots “=” -frac{1}{12}] dove i puntini a sinistra dell’uguale indicano che vanno sommati tutti i numeri naturali, mentre l’uguale è scritto tra virgolette proprio ad indicare che non si tratta di una vera uguaglianza, ma di un’uguaglianza che ha senso usando la procedura di continuazione analitica. Non ha infatti molto senso pensare che la somma di tutti i numeri naturali, infiniti e tutti positivi, sia uguale a un numero razionale e per di più negativo! In effetti la formula nasce dalla continuazione analitica proprio della funzione zeta di Riemann, e ci dice che (-frac{1}{12}) è il valore della funzione (zeta(s)) per (s=-1). Se volete saperne di più su somme di questo tipo, senza usare la continuazione analitica, potete consultare l’articolo tecnico sul blog di Terence Tao. […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_503674"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=503674#respond" data-commentid="503674" data-postid="3643" data-belowelement="comment-503674" data-respondelement="respond" data-replyto="Reply to Fake papers #1: L’uomo che scambiò la Zeta di Riemann per un polinomio | Maddmaths!" aria-label="Reply to Fake papers #1: L’uomo che scambiò la Zeta di Riemann per un polinomio | Maddmaths!">Reply</a> </div> </div> <div class="comment even thread-odd thread-alt depth-1 vcard parent" id="comment-505743"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-505743" title="Permalink to this comment">7 October, 2018 at 4:14 pm</a></p> <p class="comment-author"><strong class="fn">Jotvansh</strong></p> </div> <div class="comment-content"> <img alt='' src='https://2.gravatar.com/avatar/296957d13f7c4d6fedf1b690b04de442ca136730b3f1bf5301af39bd9b207459?s=32&d=identicon&r=PG' srcset='https://2.gravatar.com/avatar/296957d13f7c4d6fedf1b690b04de442ca136730b3f1bf5301af39bd9b207459?s=32&d=identicon&r=PG 1x, https://2.gravatar.com/avatar/296957d13f7c4d6fedf1b690b04de442ca136730b3f1bf5301af39bd9b207459?s=48&d=identicon&r=PG 1.5x, https://2.gravatar.com/avatar/296957d13f7c4d6fedf1b690b04de442ca136730b3f1bf5301af39bd9b207459?s=64&d=identicon&r=PG 2x, https://2.gravatar.com/avatar/296957d13f7c4d6fedf1b690b04de442ca136730b3f1bf5301af39bd9b207459?s=96&d=identicon&r=PG 3x, https://2.gravatar.com/avatar/296957d13f7c4d6fedf1b690b04de442ca136730b3f1bf5301af39bd9b207459?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Hi, Professor Tao </p> <p>I am a highschool student, I was really fascinated with with this theorem as it contradicts the basic principles of mathematics, so I chose this my mathematical reasacrch project, however I have to prove this summation in an less rigorous way. I have used Grandi series and Ceasaro convergent in order to use for this equation. Although, I am not really sure it would be correct to use them to manipulate the series.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_505743"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=505743#respond" data-commentid="505743" data-postid="3643" data-belowelement="comment-505743" data-respondelement="respond" data-replyto="Reply to Jotvansh" aria-label="Reply to Jotvansh">Reply</a> </div> </div> <ul class="children"> <div class="comment odd alt depth-2 vcard parent" id="comment-505751"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-505751" title="Permalink to this comment">8 October, 2018 at 2:09 am</a></p> <p class="comment-author"><strong class="fn">Ángel Méndez Rivera</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/33310ec1ed4a0024da177757b5131cd6c1d8ada587b43a77ac640d16ca3c2a5a?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>First of all, you are wrong. The theorem does not contradict the basic principles of mathematics. In fact, I question whether you can even correctly list all the basic principles of mathematics you are appealing to. My guess is that you cannot. This is not surprising, since such principles are typically never taught until you have reached a graduate level of mathematical education. In any case, the very fact that it is a theorem implies it cannot be in contradiction with the principles, because by definition, theorems are derivable only from principles. If you conclude that mathematical claim contradicts the principles, then either the claim is not a theorem, or else your understanding of the principles is incorrect.</p> <p>Second of all, Cèsaro summations are not useable for this series, because this series is outside the domain of the Cèsaro summations. The Grandi series is not linearly nor polynomially related to the Ramanujan series. You must use some other method to prove the theorem.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_505751"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=505751#respond" data-commentid="505751" data-postid="3643" data-belowelement="comment-505751" data-respondelement="respond" data-replyto="Reply to Ángel Méndez Rivera" aria-label="Reply to Ángel Méndez Rivera">Reply</a> </div> </div> <ul class="children"> <div class="comment even depth-3 vcard" id="comment-679774"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-679774" title="Permalink to this comment">16 July, 2023 at 5:37 am</a></p> <p class="comment-author"><strong class="fn"><a href="https://www.hs-augsburg.de/~mueckenh/" class="url" rel="ugc external nofollow">Wolfgang Mckenhei8m</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/fcfce52d7b52f746cc1a25e97ac9a35d3be0f10b1da7e79c78fd8ac0311e2793?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/fcfce52d7b52f746cc1a25e97ac9a35d3be0f10b1da7e79c78fd8ac0311e2793?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/fcfce52d7b52f746cc1a25e97ac9a35d3be0f10b1da7e79c78fd8ac0311e2793?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/fcfce52d7b52f746cc1a25e97ac9a35d3be0f10b1da7e79c78fd8ac0311e2793?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/fcfce52d7b52f746cc1a25e97ac9a35d3be0f10b1da7e79c78fd8ac0311e2793?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/fcfce52d7b52f746cc1a25e97ac9a35d3be0f10b1da7e79c78fd8ac0311e2793?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>A theorem can imply a contradiction if the basics are inconsistent. Here: Only geometric series with q less than 1 can be summed in a meaningful way. Further summing infinit sets obeys the same logic as summing finite sets: Adding only positive values cannot reduce the result. Finally we need not look for the reasons if the result is blatantly false like -1/12 > 1.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_679774"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=679774#respond" data-commentid="679774" data-postid="3643" data-belowelement="comment-679774" data-respondelement="respond" data-replyto="Reply to Wolfgang Mckenhei8m" aria-label="Reply to Wolfgang Mckenhei8m">Reply</a> </div> </div> </ul><!-- .children --> </ul><!-- .children --> <div class="comment odd alt thread-even depth-1 vcard" id="comment-511897"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-511897" title="Permalink to this comment">29 January, 2019 at 4:00 pm</a></p> <p class="comment-author"><strong class="fn">Jerome</strong></p> </div> <div class="comment-content"> <img alt='' src='https://1.gravatar.com/avatar/762a6b4c190a13ac37e5a8cb4bac385f8b3876487ad3d3f992b525db556a353e?s=32&d=identicon&r=PG' srcset='https://1.gravatar.com/avatar/762a6b4c190a13ac37e5a8cb4bac385f8b3876487ad3d3f992b525db556a353e?s=32&d=identicon&r=PG 1x, https://1.gravatar.com/avatar/762a6b4c190a13ac37e5a8cb4bac385f8b3876487ad3d3f992b525db556a353e?s=48&d=identicon&r=PG 1.5x, https://1.gravatar.com/avatar/762a6b4c190a13ac37e5a8cb4bac385f8b3876487ad3d3f992b525db556a353e?s=64&d=identicon&r=PG 2x, https://1.gravatar.com/avatar/762a6b4c190a13ac37e5a8cb4bac385f8b3876487ad3d3f992b525db556a353e?s=96&d=identicon&r=PG 3x, https://1.gravatar.com/avatar/762a6b4c190a13ac37e5a8cb4bac385f8b3876487ad3d3f992b525db556a353e?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>@RIvera I support your patience.<br /> A small linguistic remark. In the context of these ‘provocative’ limits (they are not a provocation at all to me.).<br /> I might be good avoid saying ‘this series has such limit’ and say this series is given such limit.<br /> In fact what matters before all is that the limit we GIVE commute with finite sums.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_511897"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=511897#respond" data-commentid="511897" data-postid="3643" data-belowelement="comment-511897" data-respondelement="respond" data-replyto="Reply to Jerome" aria-label="Reply to Jerome">Reply</a> </div> </div> <div class="comment even thread-odd thread-alt depth-1 vcard" id="comment-519024"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-519024" title="Permalink to this comment">29 July, 2019 at 4:17 am</a></p> <p class="comment-author"><strong class="fn"><a href="http://tph.tuwien.ac.at/~svozil/" class="url" rel="ugc external nofollow">Karl Svozil</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://2.gravatar.com/avatar/5f585bcf5f80d142a36a06e9feb21922b2eb6fe3c208dcc7141136cd64c4b227?s=32&d=identicon&r=PG' srcset='https://2.gravatar.com/avatar/5f585bcf5f80d142a36a06e9feb21922b2eb6fe3c208dcc7141136cd64c4b227?s=32&d=identicon&r=PG 1x, https://2.gravatar.com/avatar/5f585bcf5f80d142a36a06e9feb21922b2eb6fe3c208dcc7141136cd64c4b227?s=48&d=identicon&r=PG 1.5x, https://2.gravatar.com/avatar/5f585bcf5f80d142a36a06e9feb21922b2eb6fe3c208dcc7141136cd64c4b227?s=64&d=identicon&r=PG 2x, https://2.gravatar.com/avatar/5f585bcf5f80d142a36a06e9feb21922b2eb6fe3c208dcc7141136cd64c4b227?s=96&d=identicon&r=PG 3x, https://2.gravatar.com/avatar/5f585bcf5f80d142a36a06e9feb21922b2eb6fe3c208dcc7141136cd64c4b227?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>May formula (12) not also be interpreted as a consequence or rather an instance of Ritt’s theorem? And therefore (14) a generalization thereof?</p> <p>(Please excuse this naive question of a humble physicist; and thank you for this very nice post!)</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_519024"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=519024#respond" data-commentid="519024" data-postid="3643" data-belowelement="comment-519024" data-respondelement="respond" data-replyto="Reply to Karl Svozil" aria-label="Reply to Karl Svozil">Reply</a> </div> </div> <div class="pingback odd alt thread-even depth-1 vcard" id="comment-520815"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-520815" title="Permalink to this comment">30 August, 2019 at 11:26 pm</a></p> <p class="comment-author"><strong class="fn"><a href="https://nevinmanimala.net/2019/08/28/terence-tao-understanding-123-1-12-without-complex-analysis/" class="url" rel="ugc external nofollow">Terence Tao - Understanding 1+2+3+...=-1/12 without Complex Analysis - Nevin Manimala's Blog</a></strong></p> </div> <div class="comment-content"> <p>[…] by /u/MysteriousSeaPeoples [link] […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_520815"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=520815#respond" data-commentid="520815" data-postid="3643" data-belowelement="comment-520815" data-respondelement="respond" data-replyto="Reply to Terence Tao - Understanding 1+2+3+...=-1/12 without Complex Analysis - Nevin Manimala's Blog" aria-label="Reply to Terence Tao - Understanding 1+2+3+...=-1/12 without Complex Analysis - Nevin Manimala's Blog">Reply</a> </div> </div> <div class="comment even thread-odd thread-alt depth-1 vcard" id="comment-523638"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-523638" title="Permalink to this comment">29 September, 2019 at 4:44 am</a></p> <p class="comment-author"><strong class="fn">Michele Nardelli</strong></p> </div> <div class="comment-content"> <img alt='' src='https://1.gravatar.com/avatar/7cfd2ba4d2030eaa1910a41195e6954fec21bf97b6c7516d6b64be90e3982f43?s=32&d=identicon&r=PG' srcset='https://1.gravatar.com/avatar/7cfd2ba4d2030eaa1910a41195e6954fec21bf97b6c7516d6b64be90e3982f43?s=32&d=identicon&r=PG 1x, https://1.gravatar.com/avatar/7cfd2ba4d2030eaa1910a41195e6954fec21bf97b6c7516d6b64be90e3982f43?s=48&d=identicon&r=PG 1.5x, https://1.gravatar.com/avatar/7cfd2ba4d2030eaa1910a41195e6954fec21bf97b6c7516d6b64be90e3982f43?s=64&d=identicon&r=PG 2x, https://1.gravatar.com/avatar/7cfd2ba4d2030eaa1910a41195e6954fec21bf97b6c7516d6b64be90e3982f43?s=96&d=identicon&r=PG 3x, https://1.gravatar.com/avatar/7cfd2ba4d2030eaa1910a41195e6954fec21bf97b6c7516d6b64be90e3982f43?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Dear Prof. Tao,<br /> I deal mainly of the new possible mathematical connections, between various formulas of some sectors of theoretical physics and some formulas of specific areas of Number Theory, especially the mathematics of the Indian genius S. Ramanujan (Rogers-Ramanujan identity, mock theta functions and partition functions) and I have already obtained several interesting results from various connections with some sectors of M-Theory, Particle Physics, and black holes physics. I think it’s important to highlight that the values of any entropies (or masses), from which you can to obtain mass (or entropy), radius and temperature of a black hole, by the develop with a formula which contains two results of mock theta functions, provide ALWAYS solutions that, in my humble opinion, could be interesting and significant, as it very closed (practically almost equals) to the mathematical constant Phi (golden ratio) and the Riemann zeta function, precisely ζ(2) = ℼ2 / 6 = 1.644934… that appears very often in many sectors of string theory and Number Theory.<br /> Thank You and best regards</p> <p><a href="https://www.semanticscholar.org/paper/On-the-Hypothetical-Dark-Matter-Candidate-New-with-Nardelli-Nardelli/4fad57b020b5ffb1c9990fba326a438368d76aab" rel="nofollow ugc">https://www.semanticscholar.org/paper/On-the-Hypothetical-Dark-Matter-Candidate-New-with-Nardelli-Nardelli/4fad57b020b5ffb1c9990fba326a438368d76aab</a></p> <p><a href="https://www.semanticscholar.org/paper/Further-Mathematical-Connections-Between-the-Dark-Nardelli-Nardelli/50c23911e187ed97e207d9bf7e5552d7a900235e" rel="nofollow ugc">https://www.semanticscholar.org/paper/Further-Mathematical-Connections-Between-the-Dark-Nardelli-Nardelli/50c23911e187ed97e207d9bf7e5552d7a900235e</a></p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_523638"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=523638#respond" data-commentid="523638" data-postid="3643" data-belowelement="comment-523638" data-respondelement="respond" data-replyto="Reply to Michele Nardelli" aria-label="Reply to Michele Nardelli">Reply</a> </div> </div> <div class="pingback odd alt thread-even depth-1 vcard" id="comment-524060"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-524060" title="Permalink to this comment">4 October, 2019 at 11:39 am</a></p> <p class="comment-author"><strong class="fn"><a href="https://thismonthifoundout.wordpress.com/2019/02/25/the-sum-of-all-numbers-is-1-12/" class="url" rel="ugc external nofollow">The Sum Of All Numbers Is -1/12? – This Month I Found Out</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://thismonthifoundout.files.wordpress.com/2018/07/cropped-medicine-snake-e1532514694258.jpg?w=32' srcset='https://thismonthifoundout.files.wordpress.com/2018/07/cropped-medicine-snake-e1532514694258.jpg?w=32 1x, https://thismonthifoundout.files.wordpress.com/2018/07/cropped-medicine-snake-e1532514694258.jpg?w=48 1.5x, https://thismonthifoundout.files.wordpress.com/2018/07/cropped-medicine-snake-e1532514694258.jpg?w=64 2x, https://thismonthifoundout.files.wordpress.com/2018/07/cropped-medicine-snake-e1532514694258.jpg?w=96 3x, https://thismonthifoundout.files.wordpress.com/2018/07/cropped-medicine-snake-e1532514694258.jpg?w=128 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>[…] <a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-fun&#8230" rel="ugc">https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-fun&#8230</a>; […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_524060"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=524060#respond" data-commentid="524060" data-postid="3643" data-belowelement="comment-524060" data-respondelement="respond" data-replyto="Reply to The Sum Of All Numbers Is -1/12? – This Month I Found Out" aria-label="Reply to The Sum Of All Numbers Is -1/12? – This Month I Found Out">Reply</a> </div> </div> <div class="comment even thread-odd thread-alt depth-1 vcard" id="comment-527316"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-527316" title="Permalink to this comment">1 November, 2019 at 8:15 am</a></p> <p class="comment-author"><strong class="fn">Anonymous</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Very Nice!</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_527316"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=527316#respond" data-commentid="527316" data-postid="3643" data-belowelement="comment-527316" data-respondelement="respond" data-replyto="Reply to Anonymous" aria-label="Reply to Anonymous">Reply</a> </div> </div> <div class="pingback odd alt thread-even depth-1 vcard" id="comment-547003"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-547003" title="Permalink to this comment">19 March, 2020 at 10:40 am</a></p> <p class="comment-author"><strong class="fn"><a href="https://elkalamaras.wordpress.com/2020/03/19/about-1-12/" class="url" rel="ugc external nofollow">About -1/12 – elkalamaras</a></strong></p> </div> <div class="comment-content"> <p>[…] [10] Terrence Tao, The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, <a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-fun&#8230" rel="ugc">https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-fun&#8230</a>; […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_547003"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=547003#respond" data-commentid="547003" data-postid="3643" data-belowelement="comment-547003" data-respondelement="respond" data-replyto="Reply to About -1/12 – elkalamaras" aria-label="Reply to About -1/12 – elkalamaras">Reply</a> </div> </div> <div class="pingback even thread-odd thread-alt depth-1 vcard" id="comment-548712"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-548712" title="Permalink to this comment">27 March, 2020 at 6:41 pm</a></p> <p class="comment-author"><strong class="fn"><a href="https://mosqueeto.net/2014/01/20/there-is-a-lot-of-discussion-in-various-online-mathematical-forums-currently-about-the-interpretation-derivation/" class="url" rel="ugc external nofollow">There is a lot of discussion in various online mathematical forums currently about the interpretation, derivation,… – mosqueeto</a></strong></p> </div> <div class="comment-content"> <p>[…] https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-func… […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_548712"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=548712#respond" data-commentid="548712" data-postid="3643" data-belowelement="comment-548712" data-respondelement="respond" data-replyto="Reply to There is a lot of discussion in various online mathematical forums currently about the interpretation, derivation,… – mosqueeto" aria-label="Reply to There is a lot of discussion in various online mathematical forums currently about the interpretation, derivation,… – mosqueeto">Reply</a> </div> </div> <div class="comment odd alt thread-even depth-1 vcard" id="comment-550316"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-550316" title="Permalink to this comment">5 April, 2020 at 4:20 am</a></p> <p class="comment-author"><strong class="fn"><a href="http://myzeta.125mb.com/" class="url" rel="ugc external nofollow">isaac mor</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/c6d6677d805ee4ec3630db9b22bd3b739951c4e1706a98c26c1f052a4cd64fe8?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/c6d6677d805ee4ec3630db9b22bd3b739951c4e1706a98c26c1f052a4cd64fe8?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/c6d6677d805ee4ec3630db9b22bd3b739951c4e1706a98c26c1f052a4cd64fe8?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/c6d6677d805ee4ec3630db9b22bd3b739951c4e1706a98c26c1f052a4cd64fe8?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/c6d6677d805ee4ec3630db9b22bd3b739951c4e1706a98c26c1f052a4cd64fe8?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/c6d6677d805ee4ec3630db9b22bd3b739951c4e1706a98c26c1f052a4cd64fe8?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>terry tap please check the png image attached</p> <p>1. (-1) factprial is -1<br /> 2. there is no pole at zeta(1)</p> <p><a href="http://myzeta.125mb.com/" rel="nofollow ugc">http://myzeta.125mb.com/</a></p> <p><a href="https://drive.google.com/file/d/1rrX3_Tx1hmbAcq3VzSMLC-uIXC5jDXQ-/view" rel="nofollow ugc">https://drive.google.com/file/d/1rrX3_Tx1hmbAcq3VzSMLC-uIXC5jDXQ-/view</a></p> <p>thanks</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_550316"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=550316#respond" data-commentid="550316" data-postid="3643" data-belowelement="comment-550316" data-respondelement="respond" data-replyto="Reply to isaac mor" aria-label="Reply to isaac mor">Reply</a> </div> </div> <div class="pingback even thread-odd thread-alt depth-1 vcard" id="comment-577936"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-577936" title="Permalink to this comment">28 July, 2020 at 11:39 pm</a></p> <p class="comment-author"><strong class="fn"><a href="https://unavistacircular.wordpress.com/2020/07/29/sorpresas-en-las-sumas-infinitas-viii-revisitando-1234-1-12/" class="url" rel="ugc external nofollow">Sorpresas en las sumas infinitas (VIII) Revisitando 1+2+3+4+…=-1/12 (?) | Una vista circular</a></strong></p> </div> <div class="comment-content"> <p>[…] se puede entender el resultado con un planteamiento elemental, que yo he aprendido de un magnifico post en el blog de Terence Tao, y que no he visto en ningún otro lugar. Esto es lo que quiero comentar hoy, para ir cerrando esta […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_577936"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=577936#respond" data-commentid="577936" data-postid="3643" data-belowelement="comment-577936" data-respondelement="respond" data-replyto="Reply to Sorpresas en las sumas infinitas (VIII) Revisitando 1+2+3+4+…=-1/12 (?) | Una vista circular" aria-label="Reply to Sorpresas en las sumas infinitas (VIII) Revisitando 1+2+3+4+…=-1/12 (?) | Una vista circular">Reply</a> </div> </div> <div class="comment odd alt thread-even depth-1 vcard" id="comment-581404"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-581404" title="Permalink to this comment">11 August, 2020 at 8:51 am</a></p> <p class="comment-author"><strong class="fn"><a href="https://divergent.thinkific.com/courses/dsi-101" class="url" rel="ugc external nofollow">Francois Oger</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/ff18f371c959acde8f33f917c7982969df0c52b61afdf24601fdd5c2f4918b73?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/ff18f371c959acde8f33f917c7982969df0c52b61afdf24601fdd5c2f4918b73?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/ff18f371c959acde8f33f917c7982969df0c52b61afdf24601fdd5c2f4918b73?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/ff18f371c959acde8f33f917c7982969df0c52b61afdf24601fdd5c2f4918b73?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/ff18f371c959acde8f33f917c7982969df0c52b61afdf24601fdd5c2f4918b73?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/ff18f371c959acde8f33f917c7982969df0c52b61afdf24601fdd5c2f4918b73?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>If you are interested to learn more about divergent series and want to understand why and how 1+2+3+4+5+6+… = -1/12, I recommend the following online course:<br /> Introduction to Divergent Series of Integers<br /> <a href="https://divergent.thinkific.com/courses/dsi-101" rel="nofollow ugc">https://divergent.thinkific.com/courses/dsi-101</a></p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_581404"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=581404#respond" data-commentid="581404" data-postid="3643" data-belowelement="comment-581404" data-respondelement="respond" data-replyto="Reply to Francois Oger" aria-label="Reply to Francois Oger">Reply</a> </div> </div> <div class="comment byuser comment-author-fpmarin even thread-odd thread-alt depth-1 vcard" id="comment-599994"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-599994" title="Permalink to this comment">28 October, 2020 at 4:36 pm</a></p> <p class="comment-author"><strong class="fn"><a href="http://fpmarin.wordpress.com" class="url" rel="ugc external nofollow">fpmarin</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://2.gravatar.com/avatar/548bf061d2acd096ad39aad59369b3ce0336fcca73070e2f34084e5f0cb83b7b?s=32&d=identicon&r=PG' srcset='https://2.gravatar.com/avatar/548bf061d2acd096ad39aad59369b3ce0336fcca73070e2f34084e5f0cb83b7b?s=32&d=identicon&r=PG 1x, https://2.gravatar.com/avatar/548bf061d2acd096ad39aad59369b3ce0336fcca73070e2f34084e5f0cb83b7b?s=48&d=identicon&r=PG 1.5x, https://2.gravatar.com/avatar/548bf061d2acd096ad39aad59369b3ce0336fcca73070e2f34084e5f0cb83b7b?s=64&d=identicon&r=PG 2x, https://2.gravatar.com/avatar/548bf061d2acd096ad39aad59369b3ce0336fcca73070e2f34084e5f0cb83b7b?s=96&d=identicon&r=PG 3x, https://2.gravatar.com/avatar/548bf061d2acd096ad39aad59369b3ce0336fcca73070e2f34084e5f0cb83b7b?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Do you know an analytical continuation of the Euler Number $E_{n}$ ?. Thanks.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_599994"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=599994#respond" data-commentid="599994" data-postid="3643" data-belowelement="comment-599994" data-respondelement="respond" data-replyto="Reply to fpmarin" aria-label="Reply to fpmarin">Reply</a> </div> </div> <div class="comment odd alt thread-even depth-1 vcard" id="comment-604770"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-604770" title="Permalink to this comment">16 November, 2020 at 8:30 am</a></p> <p class="comment-author"><strong class="fn">Patrick D'Anzi</strong></p> </div> <div class="comment-content"> <img alt='' src='https://graph.facebook.com/v6.0/10208501168524523/picture?type=large&_md5=4d0b01089d3522035a77cc7e6988aad1' srcset='https://graph.facebook.com/v6.0/10208501168524523/picture?type=large&_md5=4d0b01089d3522035a77cc7e6988aad1 1x, https://graph.facebook.com/v6.0/10208501168524523/picture?type=large&_md5=4d0b01089d3522035a77cc7e6988aad1 1.5x, https://graph.facebook.com/v6.0/10208501168524523/picture?type=large&_md5=4d0b01089d3522035a77cc7e6988aad1 2x, https://graph.facebook.com/v6.0/10208501168524523/picture?type=large&_md5=4d0b01089d3522035a77cc7e6988aad1 3x, https://graph.facebook.com/v6.0/10208501168524523/picture?type=large&_md5=4d0b01089d3522035a77cc7e6988aad1 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>I have difficulty understanding how to make this statement explicit (I don’t know complex analysis very well), what I did was use the Euler-Maclaurin formula:</p> <p>\[ \int^n_m f(z)dz=\sum_{k=m}^nf(k)-\frac{f(n)-f(m)}{2}-\sum_{h=1}^{\lfloor p/2\rfloor}\frac{B_{2h}}{(2h)!}(f^{(2h-1)}(n)-f^{(2h-1)}(m))-R_p \]</p> <p>where \(R_p=\mathcal{O}(\int_m^n |f^{(p)}(z)|dz) \).</p> <p>let \( \alpha=\mathcal{Re}[s-B]$ and $g(z)=\zeta(z)\frac{N^{z-s}F(z-s)}{z-s}: \)</p> <p>\[ \int^{\alpha+i \infty}_{\alpha-i \infty} g(z)dz=\sum_{k=-\infty}^{+ \infty}g(\alpha+i k)+\mathcal{O}(\int_{\alpha-i \infty}^{\alpha+i \infty} |g^{(p)}(z)|dz) \]<br /> at this point I have doubts, I could change the integration line by choosing a \( \alpha \) that \( \forall k \) neglects \( g (\alpha + ik) \) but anyway I don’t know how to show that \( \mathcal {O} (\int_ {\alpha-i \infty} ^ {\alpha + i \infty} | g ^ {(p)} (z) | dz) = \mathcal {O} (N ^ {- B}) \)</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_604770"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=604770#respond" data-commentid="604770" data-postid="3643" data-belowelement="comment-604770" data-respondelement="respond" data-replyto="Reply to Patrick D'Anzi" aria-label="Reply to Patrick D'Anzi">Reply</a> </div> </div> <div class="comment byuser comment-author-tcjpn even thread-odd thread-alt depth-1 vcard parent" id="comment-621769"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-621769" title="Permalink to this comment">31 January, 2021 at 11:28 am</a></p> <p class="comment-author"><strong class="fn"><a href="http://tcjpn.wordpress.com" class="url" rel="ugc external nofollow">Tom Copeland</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://2.gravatar.com/avatar/ebcd7cb2c9dc4a6d300ffe4fa71394d7a45d3650c9e3c80158bfced506e823b2?s=32&d=identicon&r=PG' srcset='https://2.gravatar.com/avatar/ebcd7cb2c9dc4a6d300ffe4fa71394d7a45d3650c9e3c80158bfced506e823b2?s=32&d=identicon&r=PG 1x, https://2.gravatar.com/avatar/ebcd7cb2c9dc4a6d300ffe4fa71394d7a45d3650c9e3c80158bfced506e823b2?s=48&d=identicon&r=PG 1.5x, https://2.gravatar.com/avatar/ebcd7cb2c9dc4a6d300ffe4fa71394d7a45d3650c9e3c80158bfced506e823b2?s=64&d=identicon&r=PG 2x, https://2.gravatar.com/avatar/ebcd7cb2c9dc4a6d300ffe4fa71394d7a45d3650c9e3c80158bfced506e823b2?s=96&d=identicon&r=PG 3x, https://2.gravatar.com/avatar/ebcd7cb2c9dc4a6d300ffe4fa71394d7a45d3650c9e3c80158bfced506e823b2?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>To see the beauty of Ramanujan’s use of divergent series, as first recognized by Hardy, see my MathOverflow answer and the comments attached to it (<a href="https://mathoverflow.net/questions/79868/what-does-mellin-inversion-really-mean/79925#79925" rel="nofollow ugc">https://mathoverflow.net/questions/79868/what-does-mellin-inversion-really-mean/79925#79925</a>). Hardy was predisposed to see this (took him overnight though) since he basically advocated in one of his early papers what I call the Hardy heuristic “When in doubt, interchange operations” similar to Feynman’s “When in doubt, integrate by parts,” the basis to the theory of distributions. (Hardy later gave rigorous conditions under which Ramanujan’s Master Formula is valid.)</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_621769"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=621769#respond" data-commentid="621769" data-postid="3643" data-belowelement="comment-621769" data-respondelement="respond" data-replyto="Reply to Tom Copeland" aria-label="Reply to Tom Copeland">Reply</a> </div> </div> <ul class="children"> <div class="comment odd alt depth-2 vcard parent" id="comment-621773"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-621773" title="Permalink to this comment">31 January, 2021 at 12:47 pm</a></p> <p class="comment-author"><strong class="fn">Michele</strong></p> </div> <div class="comment-content"> <img alt='' src='https://1.gravatar.com/avatar/7cfd2ba4d2030eaa1910a41195e6954fec21bf97b6c7516d6b64be90e3982f43?s=32&d=identicon&r=PG' srcset='https://1.gravatar.com/avatar/7cfd2ba4d2030eaa1910a41195e6954fec21bf97b6c7516d6b64be90e3982f43?s=32&d=identicon&r=PG 1x, https://1.gravatar.com/avatar/7cfd2ba4d2030eaa1910a41195e6954fec21bf97b6c7516d6b64be90e3982f43?s=48&d=identicon&r=PG 1.5x, https://1.gravatar.com/avatar/7cfd2ba4d2030eaa1910a41195e6954fec21bf97b6c7516d6b64be90e3982f43?s=64&d=identicon&r=PG 2x, https://1.gravatar.com/avatar/7cfd2ba4d2030eaa1910a41195e6954fec21bf97b6c7516d6b64be90e3982f43?s=96&d=identicon&r=PG 3x, https://1.gravatar.com/avatar/7cfd2ba4d2030eaa1910a41195e6954fec21bf97b6c7516d6b64be90e3982f43?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Great Srinivasa Ramanujan! is my source of inspiration and was a genius, as Littlewood said, comparable to a Jacobi or an Euler</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_621773"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=621773#respond" data-commentid="621773" data-postid="3643" data-belowelement="comment-621773" data-respondelement="respond" data-replyto="Reply to Michele" aria-label="Reply to Michele">Reply</a> </div> </div> <ul class="children"> <div class="comment byuser comment-author-tcjpn even depth-3 vcard" id="comment-621774"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-621774" title="Permalink to this comment">31 January, 2021 at 1:03 pm</a></p> <p class="comment-author"><strong class="fn"><a href="http://tcjpn.wordpress.com" class="url" rel="ugc external nofollow">Tom Copeland</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://2.gravatar.com/avatar/ebcd7cb2c9dc4a6d300ffe4fa71394d7a45d3650c9e3c80158bfced506e823b2?s=32&d=identicon&r=PG' srcset='https://2.gravatar.com/avatar/ebcd7cb2c9dc4a6d300ffe4fa71394d7a45d3650c9e3c80158bfced506e823b2?s=32&d=identicon&r=PG 1x, https://2.gravatar.com/avatar/ebcd7cb2c9dc4a6d300ffe4fa71394d7a45d3650c9e3c80158bfced506e823b2?s=48&d=identicon&r=PG 1.5x, https://2.gravatar.com/avatar/ebcd7cb2c9dc4a6d300ffe4fa71394d7a45d3650c9e3c80158bfced506e823b2?s=64&d=identicon&r=PG 2x, https://2.gravatar.com/avatar/ebcd7cb2c9dc4a6d300ffe4fa71394d7a45d3650c9e3c80158bfced506e823b2?s=96&d=identicon&r=PG 3x, https://2.gravatar.com/avatar/ebcd7cb2c9dc4a6d300ffe4fa71394d7a45d3650c9e3c80158bfced506e823b2?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Hardy rated himself a C mathematician compared to Ramanujan as an A. (That puts me off the alphabet. I don’t use the term genius– in some sense it marginalizes the passion, the diligence and dedication, even obsession, of the masters.)</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_621774"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=621774#respond" data-commentid="621774" data-postid="3643" data-belowelement="comment-621774" data-respondelement="respond" data-replyto="Reply to Tom Copeland" aria-label="Reply to Tom Copeland">Reply</a> </div> </div> </ul><!-- .children --> </ul><!-- .children --> <div class="pingback odd alt thread-even depth-1 vcard" id="comment-622256"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-622256" title="Permalink to this comment">12 February, 2021 at 9:38 am</a></p> <p class="comment-author"><strong class="fn"><a href="https://terrytao.wordpress.com/2021/02/12/246b-notes-4-the-riemann-zeta-function-and-the-prime-number-theorem/" class="url" rel="ugc">246B, Notes 4: The Riemann zeta function and the prime number theorem | What's new</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://secure.gravatar.com/blavatar/bd4bda4207561b6998f10dec44b570f04ff4072b20f89162d525b186dfca3e49?s=32' srcset='https://secure.gravatar.com/blavatar/bd4bda4207561b6998f10dec44b570f04ff4072b20f89162d525b186dfca3e49?s=32 1x, https://secure.gravatar.com/blavatar/bd4bda4207561b6998f10dec44b570f04ff4072b20f89162d525b186dfca3e49?s=48 1.5x, https://secure.gravatar.com/blavatar/bd4bda4207561b6998f10dec44b570f04ff4072b20f89162d525b186dfca3e49?s=64 2x, https://secure.gravatar.com/blavatar/bd4bda4207561b6998f10dec44b570f04ff4072b20f89162d525b186dfca3e49?s=96 3x, https://secure.gravatar.com/blavatar/bd4bda4207561b6998f10dec44b570f04ff4072b20f89162d525b186dfca3e49?s=128 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>[…] has to be interpreted in a suitable non-classical sense in order for it to be rigorous (see this previous blog post for further […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_622256"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=622256#respond" data-commentid="622256" data-postid="3643" data-belowelement="comment-622256" data-respondelement="respond" data-replyto="Reply to 246B, Notes 4: The Riemann zeta function and the prime number theorem | What's new" aria-label="Reply to 246B, Notes 4: The Riemann zeta function and the prime number theorem | What's new">Reply</a> </div> </div> <div class="pingback even thread-odd thread-alt depth-1 vcard" id="comment-633951"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-633951" title="Permalink to this comment">11 September, 2021 at 4:28 pm</a></p> <p class="comment-author"><strong class="fn"><a href="https://derivative.blog/links/" class="url" rel="ugc external nofollow">Links | Derivative Blog</a></strong></p> </div> <div class="comment-content"> <p>[…] The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic conti… […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_633951"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=633951#respond" data-commentid="633951" data-postid="3643" data-belowelement="comment-633951" data-respondelement="respond" data-replyto="Reply to Links | Derivative Blog" aria-label="Reply to Links | Derivative Blog">Reply</a> </div> </div> <div class="comment odd alt thread-even depth-1 vcard" id="comment-637652"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-637652" title="Permalink to this comment">28 November, 2021 at 11:27 am</a></p> <p class="comment-author"><strong class="fn">Ted</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/9c6baf138c72142bc99c2322d4bbaf9727f29f57727afc68b4f5527e32d7bc64?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/9c6baf138c72142bc99c2322d4bbaf9727f29f57727afc68b4f5527e32d7bc64?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/9c6baf138c72142bc99c2322d4bbaf9727f29f57727afc68b4f5527e32d7bc64?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/9c6baf138c72142bc99c2322d4bbaf9727f29f57727afc68b4f5527e32d7bc64?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/9c6baf138c72142bc99c2322d4bbaf9727f29f57727afc68b4f5527e32d7bc64?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/9c6baf138c72142bc99c2322d4bbaf9727f29f57727afc68b4f5527e32d7bc64?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>I believe that you need to strengthen your initial definition of a “cutoff function” to require that <img src="https://s0.wp.com/latex.php?latex=%5Ceta%28x%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Ceta%28x%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Ceta%28x%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\eta(x)" class="latex" /> be (right-)continuous at <img src="https://s0.wp.com/latex.php?latex=x+%3D+0&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=x+%3D+0&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=x+%3D+0&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="x = 0" class="latex" />. Your current initial definition only requires that <img src="https://s0.wp.com/latex.php?latex=%5Ceta%280%29+%3D+1&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Ceta%280%29+%3D+1&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Ceta%280%29+%3D+1&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\eta(0) = 1" class="latex" />, but this is not enough to imply your later claim that “<img src="https://s0.wp.com/latex.php?latex=%5Ceta%28n%2FN%29+%5Cto+1&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Ceta%28n%2FN%29+%5Cto+1&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Ceta%28n%2FN%29+%5Cto+1&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\eta(n/N) \to 1" class="latex" /> pointwise as <img src="https://s0.wp.com/latex.php?latex=N+%5Cto+%5Cinfty&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=N+%5Cto+%5Cinfty&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=N+%5Cto+%5Cinfty&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="N \to \infty" class="latex" />“. (Beginning with the following sentence, you consistently specify that <img src="https://s0.wp.com/latex.php?latex=%5Ceta&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Ceta&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Ceta&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\eta" class="latex" /> is (at least) continuous.)</p> <p><i>[Corrected, thanks – T.]</i></p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_637652"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=637652#respond" data-commentid="637652" data-postid="3643" data-belowelement="comment-637652" data-respondelement="respond" data-replyto="Reply to Ted" aria-label="Reply to Ted">Reply</a> </div> </div> <div class="comment even thread-odd thread-alt depth-1 vcard parent" id="comment-639574"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-639574" title="Permalink to this comment">21 January, 2022 at 1:28 am</a></p> <p class="comment-author"><strong class="fn">Amos Kipngetich Korir</strong></p> </div> <div class="comment-content"> <img alt='' src='https://2.gravatar.com/avatar/8cd8840964248c262855cd99a00a47e31dbd6481124220255a77eb5d34c7b6b4?s=32&d=identicon&r=PG' srcset='https://2.gravatar.com/avatar/8cd8840964248c262855cd99a00a47e31dbd6481124220255a77eb5d34c7b6b4?s=32&d=identicon&r=PG 1x, https://2.gravatar.com/avatar/8cd8840964248c262855cd99a00a47e31dbd6481124220255a77eb5d34c7b6b4?s=48&d=identicon&r=PG 1.5x, https://2.gravatar.com/avatar/8cd8840964248c262855cd99a00a47e31dbd6481124220255a77eb5d34c7b6b4?s=64&d=identicon&r=PG 2x, https://2.gravatar.com/avatar/8cd8840964248c262855cd99a00a47e31dbd6481124220255a77eb5d34c7b6b4?s=96&d=identicon&r=PG 3x, https://2.gravatar.com/avatar/8cd8840964248c262855cd99a00a47e31dbd6481124220255a77eb5d34c7b6b4?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Here is interesting result which one can use to come up with Euler -Maclaurin formula.I found out that 1/(1+X) is convergent for X>1 if we change the sign of exponent in the famous infinite series a(its Taylor series)to negative and this can be clearly be proven and we can still use the same coefficients.<br /> 1/(1+X)=(x^-1) – (x^-2) + (x^-3) -……..<br /> And this is true for other values like<br /> 1/(1+X)^2…</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_639574"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=639574#respond" data-commentid="639574" data-postid="3643" data-belowelement="comment-639574" data-respondelement="respond" data-replyto="Reply to Amos Kipngetich Korir" aria-label="Reply to Amos Kipngetich Korir">Reply</a> </div> </div> <ul class="children"> <div class="comment odd alt depth-2 vcard" id="comment-639582"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-639582" title="Permalink to this comment">21 January, 2022 at 1:52 pm</a></p> <p class="comment-author"><strong class="fn">Anonymous</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>This is true for any rational function <img src="https://s0.wp.com/latex.php?latex=f&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=f&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=f&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="f" class="latex" /> which satisfy the functional equation <img src="https://s0.wp.com/latex.php?latex=f%28x%5E%7B-1%7D%29+%3D+f%28x%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=f%28x%5E%7B-1%7D%29+%3D+f%28x%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=f%28x%5E%7B-1%7D%29+%3D+f%28x%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="f(x^{-1}) = f(x)" class="latex" /> – which is satisfied by the above examples.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_639582"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=639582#respond" data-commentid="639582" data-postid="3643" data-belowelement="comment-639582" data-respondelement="respond" data-replyto="Reply to Anonymous" aria-label="Reply to Anonymous">Reply</a> </div> </div> </ul><!-- .children --> <div class="pingback even thread-even depth-1 vcard" id="comment-644160"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-644160" title="Permalink to this comment">30 May, 2022 at 6:54 am</a></p> <p class="comment-author"><strong class="fn"><a href="https://sandyekahana.wordpress.com/2022/05/30/1234/" class="url" rel="ugc external nofollow">1+2+3+4+… = ? | Buah Pikir</a></strong></p> </div> <div class="comment-content"> <p>[…] blog pribadinya, Terry Tao menulis bahwa kita perlu melakukan regularisasi. Makhluk apa itu regularisasi? Saya […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_644160"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=644160#respond" data-commentid="644160" data-postid="3643" data-belowelement="comment-644160" data-respondelement="respond" data-replyto="Reply to 1+2+3+4+… = ? | Buah Pikir" aria-label="Reply to 1+2+3+4+… = ? | Buah Pikir">Reply</a> </div> </div> <div class="comment odd alt thread-odd thread-alt depth-1 vcard parent" id="comment-649520"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-649520" title="Permalink to this comment">21 July, 2022 at 11:04 am</a></p> <p class="comment-author"><strong class="fn">CK</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/956a8777c38b34eae8447779dadd578af69efd5c97d47dbac3391bc2b73c9802?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/956a8777c38b34eae8447779dadd578af69efd5c97d47dbac3391bc2b73c9802?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/956a8777c38b34eae8447779dadd578af69efd5c97d47dbac3391bc2b73c9802?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/956a8777c38b34eae8447779dadd578af69efd5c97d47dbac3391bc2b73c9802?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/956a8777c38b34eae8447779dadd578af69efd5c97d47dbac3391bc2b73c9802?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/956a8777c38b34eae8447779dadd578af69efd5c97d47dbac3391bc2b73c9802?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>In formula (23) you state: “Applying this with <img src="https://s0.wp.com/latex.php?latex=N&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=N&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=N&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="N" class="latex" /> equal to a power of two and summing the telescoping series, one concludes that (24)”. If we let N -> 2^k and sum from k=N to k=infinity, then the telescoping sum on the RHS contains a term infinity^(1-s) which is undefined for Re(s)<1 ?</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_649520"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=649520#respond" data-commentid="649520" data-postid="3643" data-belowelement="comment-649520" data-respondelement="respond" data-replyto="Reply to CK" aria-label="Reply to CK">Reply</a> </div> </div> <ul class="children"> <div class="comment byuser comment-author-teorth bypostauthor even depth-2 vcard" id="comment-649526"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-649526" title="Permalink to this comment">21 July, 2022 at 3:33 pm</a></p> <p class="comment-author"><strong class="fn"><a href="http://www.math.ucla.edu/~tao" class="url" rel="ugc external nofollow">Terence Tao</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=32&d=identicon&r=PG' srcset='https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=32&d=identicon&r=PG 1x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=48&d=identicon&r=PG 1.5x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=64&d=identicon&r=PG 2x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=96&d=identicon&r=PG 3x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Summing the telescoping series on a finite range $\sum_{k=k_0}^{k_1} \dots$ will show that <img src="https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%7B2%5Ek%7D+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2F2%5Ek%29+-+C_%7B%5Ceta%2C-s%7D+%282%5Ek%29%5E%7B1-s%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%7B2%5Ek%7D+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2F2%5Ek%29+-+C_%7B%5Ceta%2C-s%7D+%282%5Ek%29%5E%7B1-s%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%7B2%5Ek%7D+%5Cfrac%7B1%7D%7Bn%5Es%7D+%5Ceta%28n%2F2%5Ek%29+-+C_%7B%5Ceta%2C-s%7D+%282%5Ek%29%5E%7B1-s%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\sum_{n=1}^{2^k} \frac{1}{n^s} \eta(n/2^k) - C_{\eta,-s} (2^k)^{1-s}" class="latex" /> is a Cauchy sequence converging to some limit <img src="https://s0.wp.com/latex.php?latex=%5Czeta%28s%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Czeta%28s%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Czeta%28s%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\zeta(s)" class="latex" />, and then one can obtain the claim. (See also Lemma 5 of <a HREF="https://terrytao.wordpress.com/2014/11/23/254a-notes-1-elementary-multiplicative-number-theory/" rel="nofollow ugc">this blog post</a> for an abstraction of this argument.)</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_649526"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=649526#respond" data-commentid="649526" data-postid="3643" data-belowelement="comment-649526" data-respondelement="respond" data-replyto="Reply to Terence Tao" aria-label="Reply to Terence Tao">Reply</a> </div> </div> </ul><!-- .children --> <div class="pingback odd alt thread-even depth-1 vcard" id="comment-670299"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-670299" title="Permalink to this comment">13 December, 2022 at 11:41 pm</a></p> <p class="comment-author"><strong class="fn"><a href="https://forum.studentb.eu/index.php/2022/12/14/asymptotics-of-smoothed-sums-from-zeta-regularization/" class="url" rel="ugc external nofollow">Asymptotics of smoothed sums from zeta regularization – Mathematics – Forum</a></strong></p> </div> <div class="comment-content"> <p>[…] this post, Terry Tao gives exactly what I'm after but for a different notion of zeta-function regularization. […]</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_670299"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=670299#respond" data-commentid="670299" data-postid="3643" data-belowelement="comment-670299" data-respondelement="respond" data-replyto="Reply to Asymptotics of smoothed sums from zeta regularization – Mathematics – Forum" aria-label="Reply to Asymptotics of smoothed sums from zeta regularization – Mathematics – Forum">Reply</a> </div> </div> <div class="comment even thread-odd thread-alt depth-1 vcard parent" id="comment-677507"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-677507" title="Permalink to this comment">14 April, 2023 at 4:06 pm</a></p> <p class="comment-author"><strong class="fn">Bo Tielman</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/642a63b87ae562f7521f7165859292bc671026ca0d010bd9f19e4ba2323ae794?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/642a63b87ae562f7521f7165859292bc671026ca0d010bd9f19e4ba2323ae794?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/642a63b87ae562f7521f7165859292bc671026ca0d010bd9f19e4ba2323ae794?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/642a63b87ae562f7521f7165859292bc671026ca0d010bd9f19e4ba2323ae794?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/642a63b87ae562f7521f7165859292bc671026ca0d010bd9f19e4ba2323ae794?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/642a63b87ae562f7521f7165859292bc671026ca0d010bd9f19e4ba2323ae794?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Like you said, we normally define infinite sums with a limit of the partial sums, and you modified the summands by including this smoothing function eta. In order to define a limit, you need a topology. Is it possible to change (or extend) the usual Euclidian topopoly on \mathbb{R} such that can still define our infinite sums the normal way (so without the eta), and that also agrees with the values of the analytic continuation, that also doesn’t mess everything we already had for normal converging infinite sums.</p> <p>Does such a topology exist and if so, is it metrizable? </p> <p>With kind regards,<br /> Bo Tielman</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_677507"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=677507#respond" data-commentid="677507" data-postid="3643" data-belowelement="comment-677507" data-respondelement="respond" data-replyto="Reply to Bo Tielman" aria-label="Reply to Bo Tielman">Reply</a> </div> </div> <ul class="children"> <div class="comment byuser comment-author-teorth bypostauthor odd alt depth-2 vcard" id="comment-677552"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-677552" title="Permalink to this comment">17 April, 2023 at 8:03 am</a></p> <p class="comment-author"><strong class="fn"><a href="http://www.math.ucla.edu/~tao" class="url" rel="ugc external nofollow">Terence Tao</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=32&d=identicon&r=PG' srcset='https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=32&d=identicon&r=PG 1x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=48&d=identicon&r=PG 1.5x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=64&d=identicon&r=PG 2x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=96&d=identicon&r=PG 3x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Most infinite summation methods are not shift invariant, for instance <img src="https://s0.wp.com/latex.php?latex=0+%2B+1+%2B+2+%2B+%5Cdots+%3D+5%2F12&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=0+%2B+1+%2B+2+%2B+%5Cdots+%3D+5%2F12&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=0+%2B+1+%2B+2+%2B+%5Cdots+%3D+5%2F12&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="0 + 1 + 2 + \dots = 5/12" class="latex" /> evaluates to a different sum than <img src="https://s0.wp.com/latex.php?latex=1+%2B+2+%2B+3+%2B+%5Cdots+%3D+-1%2F12&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=1+%2B+2+%2B+3+%2B+%5Cdots+%3D+-1%2F12&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=1+%2B+2+%2B+3+%2B+%5Cdots+%3D+-1%2F12&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="1 + 2 + 3 + \dots = -1/12" class="latex" />, so one cannot interpret these assignments of values to infinite sums as a limit of partial sums in any reasonable (e.g., Hausdorff) topology.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_677552"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=677552#respond" data-commentid="677552" data-postid="3643" data-belowelement="comment-677552" data-respondelement="respond" data-replyto="Reply to Terence Tao" aria-label="Reply to Terence Tao">Reply</a> </div> </div> </ul><!-- .children --> <div class="comment even thread-even depth-1 vcard" id="comment-682639"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-682639" title="Permalink to this comment">19 December, 2023 at 10:52 am</a></p> <p class="comment-author"><strong class="fn">Anonymous</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>I have a strange sensation that this summation may be related to the renormalization procedure in physics. That one bothers Dirac as a deep flaw in the foundation of quantum field theory.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_682639"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=682639#respond" data-commentid="682639" data-postid="3643" data-belowelement="comment-682639" data-respondelement="respond" data-replyto="Reply to Anonymous" aria-label="Reply to Anonymous">Reply</a> </div> </div> <div class="comment odd alt thread-odd thread-alt depth-1 vcard parent" id="comment-682647"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-682647" title="Permalink to this comment">20 December, 2023 at 3:06 pm</a></p> <p class="comment-author"><strong class="fn">Anonymous</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>A classic post! I’m amazed that cutoff functions like this don’t seem to be a standard topic in the context of divergent series. As shown here they often seem very mathematically natural, and in physics contexts I think they are the most natural class of summation methods.</p> <p>One question I couldn’t immediately see the answer to, though: are they *consistent*, in the usual sense of summation methods? That is, if two functions $\eta$ and $\tilde{\eta}$ can both give a finite answer to the smoothed sum, are those answers necessarily equal? (If no, can some reasonable conditions on $\eta$ or $a_n$ make them consistent?) </p> <p>Asked here too: <a href="https://mathoverflow.net/questions/460676/consistency-results-for-divergent-series-summed-with-smooth-cutoff-functions" rel="nofollow ugc">https://mathoverflow.net/questions/460676/consistency-results-for-divergent-series-summed-with-smooth-cutoff-functions</a></p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_682647"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=682647#respond" data-commentid="682647" data-postid="3643" data-belowelement="comment-682647" data-respondelement="respond" data-replyto="Reply to Anonymous" aria-label="Reply to Anonymous">Reply</a> </div> </div> <ul class="children"> <div class="comment byuser comment-author-teorth bypostauthor even depth-2 vcard" id="comment-682669"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-682669" title="Permalink to this comment">26 December, 2023 at 4:30 pm</a></p> <p class="comment-author"><strong class="fn"><a href="http://www.math.ucla.edu/~tao" class="url" rel="ugc external nofollow">Terence Tao</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=32&d=identicon&r=PG' srcset='https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=32&d=identicon&r=PG 1x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=48&d=identicon&r=PG 1.5x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=64&d=identicon&r=PG 2x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=96&d=identicon&r=PG 3x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>For functions of polynomial growth one can usually use the Euler-Maclaurin formula (22) to show that the cutoff <img src="https://s0.wp.com/latex.php?latex=%5Ceta&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Ceta&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Ceta&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\eta" class="latex" /> only affects higher order terms and not the constant term. However for functions of faster than polynomial growth, the situation is more delicate and one would have to analyze the divergent series on a case-by-case basis. I think in some cases, one cannot work with arbitrary cutoff functions, but instead turn to special cutoffs like the one used in zeta function regularization in order to have a good theory.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_682669"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=682669#respond" data-commentid="682669" data-postid="3643" data-belowelement="comment-682669" data-respondelement="respond" data-replyto="Reply to Terence Tao" aria-label="Reply to Terence Tao">Reply</a> </div> </div> </ul><!-- .children --> <div class="comment byuser comment-author-sparshsriva odd alt thread-even depth-1 vcard parent" id="comment-682730"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-682730" title="Permalink to this comment">6 January, 2024 at 10:15 pm</a></p> <p class="comment-author"><strong class="fn"><a href="http://sparshsriva.wordpress.com" class="url" rel="ugc external nofollow">Sparsh Srivastava</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=32&d=identicon&r=PG' srcset='https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=32&d=identicon&r=PG 1x, https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=48&d=identicon&r=PG 1.5x, https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=64&d=identicon&r=PG 2x, https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=96&d=identicon&r=PG 3x, https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>You made a mistake. Equation 8 is incorrect, you added each term in the series pointwise, which is a different operation than adding 2 infinite series. Also consider pointwise multiplication of series. Gonna test these operations out on ramunajans identities now. I’m considering the sum of the reciprocal powers of odds added with the sum of the reciprocal powers of evens because it gives zeta(s)-1 when re(s)>1. Also looking at the pairwise addition case of ramunajans first theorem on summations of series and its generalization now. Very interesting results. Also you’ve read his handwritten notes right? He uses +&c instead of ellipses. I wonder what the significance of that is.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_682730"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=682730#respond" data-commentid="682730" data-postid="3643" data-belowelement="comment-682730" data-respondelement="respond" data-replyto="Reply to Sparsh Srivastava" aria-label="Reply to Sparsh Srivastava">Reply</a> </div> </div> <ul class="children"> <div class="comment byuser comment-author-sparshsriva even depth-2 vcard" id="comment-682731"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-682731" title="Permalink to this comment">6 January, 2024 at 10:19 pm</a></p> <p class="comment-author"><strong class="fn"><a href="http://sparshsriva.wordpress.com" class="url" rel="ugc external nofollow">Sparsh Srivastava</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=32&d=identicon&r=PG' srcset='https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=32&d=identicon&r=PG 1x, https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=48&d=identicon&r=PG 1.5x, https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=64&d=identicon&r=PG 2x, https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=96&d=identicon&r=PG 3x, https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Also sometimes he gives 3 terms and in other cases 4. I’m sure there’s some significance to how many terms he gives. I think he’s giving the minimum number of terms needed to continue the pattern to infinity.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_682731"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=682731#respond" data-commentid="682731" data-postid="3643" data-belowelement="comment-682731" data-respondelement="respond" data-replyto="Reply to Sparsh Srivastava" aria-label="Reply to Sparsh Srivastava">Reply</a> </div> </div> <div class="comment byuser comment-author-sparshsriva odd alt depth-2 vcard" id="comment-682732"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-682732" title="Permalink to this comment">6 January, 2024 at 10:28 pm</a></p> <p class="comment-author"><strong class="fn"><a href="http://sparshsriva.wordpress.com" class="url" rel="ugc external nofollow">Sparsh Srivastava</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=32&d=identicon&r=PG' srcset='https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=32&d=identicon&r=PG 1x, https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=48&d=identicon&r=PG 1.5x, https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=64&d=identicon&r=PG 2x, https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=96&d=identicon&r=PG 3x, https://1.gravatar.com/avatar/a5995db5a30e27dd0841b10389c6e050df0ccf0f6f4e8f29a4763e63141761cc?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Also x2 have you read the Aryabhatiyam by Aryabhata? I’m learning Sanskrit now using ai and I had it translate a bit and I read it, some parallels between it and ramunajan’s work. I can’t verify if he knew Sanskrit and would have read this text, but since its a classic Indian mathematical text I’m thinking the information flowed to him one way or another.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_682732"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=682732#respond" data-commentid="682732" data-postid="3643" data-belowelement="comment-682732" data-respondelement="respond" data-replyto="Reply to Sparsh Srivastava" aria-label="Reply to Sparsh Srivastava">Reply</a> </div> </div> </ul><!-- .children --> <div class="comment even thread-odd thread-alt depth-1 vcard" id="comment-684439"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-684439" title="Permalink to this comment">19 February, 2024 at 10:01 am</a></p> <p class="comment-author"><strong class="fn">Anonymous</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Related:</p> <div class="embed-youtube"><iframe title="Does -1/12 Protect Us From Infinity? - Numberphile" width="490" height="276" src="https://www.youtube.com/embed/beakj767uG4?feature=oembed" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe></div> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_684439"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=684439#respond" data-commentid="684439" data-postid="3643" data-belowelement="comment-684439" data-respondelement="respond" data-replyto="Reply to Anonymous" aria-label="Reply to Anonymous">Reply</a> </div> </div> <div class="comment odd alt thread-even depth-1 vcard parent" id="comment-684452"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-684452" title="Permalink to this comment">24 February, 2024 at 7:55 am</a></p> <p class="comment-author"><strong class="fn">Anonymous</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Is there a way of deriving the sum of a series from the sums of related series. For example is the sum of the natural numbers related in some way to the individual sums of the odd number and even numbers?</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_684452"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=684452#respond" data-commentid="684452" data-postid="3643" data-belowelement="comment-684452" data-respondelement="respond" data-replyto="Reply to Anonymous" aria-label="Reply to Anonymous">Reply</a> </div> </div> <ul class="children"> <div class="comment byuser comment-author-teorth bypostauthor even depth-2 vcard parent" id="comment-684456"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-684456" title="Permalink to this comment">24 February, 2024 at 1:31 pm</a></p> <p class="comment-author"><strong class="fn"><a href="http://www.math.ucla.edu/~tao" class="url" rel="ugc external nofollow">Terence Tao</a></strong></p> </div> <div class="comment-content"> <img alt='' src='https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=32&d=identicon&r=PG' srcset='https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=32&d=identicon&r=PG 1x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=48&d=identicon&r=PG 1.5x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=64&d=identicon&r=PG 2x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=96&d=identicon&r=PG 3x, https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>There are some relations, but one needs to proceed with care as there are several pitfalls. Firstly, divergent summation methods tend to be linear in the summands (provided that the sums actually can be interpreted); and hence <img src="https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\sum_{n=1}^\infty f(n)" class="latex" /> can be expressed as the sum of <img src="https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29+1_%7B2%7Cn%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29+1_%7B2%7Cn%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29+1_%7B2%7Cn%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\sum_{n=1}^\infty f(n) 1_{2|n}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29+1_%7B2%7Cn-1%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29+1_%7B2%7Cn-1%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29+1_%7B2%7Cn-1%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\sum_{n=1}^\infty f(n) 1_{2|n-1}" class="latex" /> if all series here are interpretable by the same method. Homogeneous change of variables such as <img src="https://s0.wp.com/latex.php?latex=n+%5Cmapsto+2n&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=n+%5Cmapsto+2n&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=n+%5Cmapsto+2n&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="n \mapsto 2n" class="latex" /> also are typically okay, so the sum <img src="https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29+1_%7B2%7Cn%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29+1_%7B2%7Cn%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29+1_%7B2%7Cn%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\sum_{n=1}^\infty f(n) 1_{2|n}" class="latex" /> can be interpreted as <img src="https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%282n%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%282n%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%282n%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\sum_{n=1}^\infty f(2n)" class="latex" />. However, it is not always the case that inhomogeneous change of variables formulae such as <img src="https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29+1_%7B2%7Cn-1%7D+%3D+&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29+1_%7B2%7Cn-1%7D+%3D+&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Csum_%7Bn%3D1%7D%5E%5Cinfty+f%28n%29+1_%7B2%7Cn-1%7D+%3D+&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\sum_{n=1}^\infty f(n) 1_{2|n-1} = " class="latex" />latex \sum_{n=1}^\infty f(2n-1)$ are valid (the shift by <img src="https://s0.wp.com/latex.php?latex=-1&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=-1&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=-1&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="-1" class="latex" /> can mess with the cutoff <img src="https://s0.wp.com/latex.php?latex=%5Ceta%28n%2FN%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Ceta%28n%2FN%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Ceta%28n%2FN%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\eta(n/N)" class="latex" /> in an undesirable fashion). Unless one knows exactly what one is doing, I would avoid blindly manipulating infinite divergent series, and work instead with the rigorous definitions of whatever summation method one is choosing to use, and manipulate those definitions from first principles.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_684456"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=684456#respond" data-commentid="684456" data-postid="3643" data-belowelement="comment-684456" data-respondelement="respond" data-replyto="Reply to Terence Tao" aria-label="Reply to Terence Tao">Reply</a> </div> </div> <ul class="children"> <div class="comment odd alt depth-3 vcard" id="comment-684459"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-684459" title="Permalink to this comment">25 February, 2024 at 3:43 am</a></p> <p class="comment-author"><strong class="fn">Anonymous</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Thanks for replying, it’s fascinating. I came to you via your recent Numberphile appearance.</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_684459"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=684459#respond" data-commentid="684459" data-postid="3643" data-belowelement="comment-684459" data-respondelement="respond" data-replyto="Reply to Anonymous" aria-label="Reply to Anonymous">Reply</a> </div> </div> </ul><!-- .children --> </ul><!-- .children --> <div class="comment byuser comment-author-orgesleka even thread-odd thread-alt depth-1 vcard" id="comment-684460"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-684460" title="Permalink to this comment">25 February, 2024 at 9:51 am</a></p> <p class="comment-author"><strong class="fn">orgesleka</strong></p> </div> <div class="comment-content"> <img alt='' src='https://2.gravatar.com/avatar/2be8ec67a43b7d6f663e629490713e2fc889aadbc5504a8a02dc66111b58896b?s=32&d=identicon&r=PG' srcset='https://2.gravatar.com/avatar/2be8ec67a43b7d6f663e629490713e2fc889aadbc5504a8a02dc66111b58896b?s=32&d=identicon&r=PG 1x, https://2.gravatar.com/avatar/2be8ec67a43b7d6f663e629490713e2fc889aadbc5504a8a02dc66111b58896b?s=48&d=identicon&r=PG 1.5x, https://2.gravatar.com/avatar/2be8ec67a43b7d6f663e629490713e2fc889aadbc5504a8a02dc66111b58896b?s=64&d=identicon&r=PG 2x, https://2.gravatar.com/avatar/2be8ec67a43b7d6f663e629490713e2fc889aadbc5504a8a02dc66111b58896b?s=96&d=identicon&r=PG 3x, https://2.gravatar.com/avatar/2be8ec67a43b7d6f663e629490713e2fc889aadbc5504a8a02dc66111b58896b?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Dear Terence Tao, </p> <p>I am not sure where to ask a question relating the Liouville function and an infinite dimensional lattice, so I thought to ask it here: </p> <p>Is this infinite dimensional lattice known in literature:<br /> <a href="https://mathoverflow.net/questions/465877/infinite-dimensional-lattice-for-integers-and-the-riemann-hypothesis" rel="nofollow ugc">https://mathoverflow.net/questions/465877/infinite-dimensional-lattice-for-integers-and-the-riemann-hypothesis</a></p> <p>Kind regards from Limburg in Germany,<br /> Orges Leka</p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_684460"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=684460#respond" data-commentid="684460" data-postid="3643" data-belowelement="comment-684460" data-respondelement="respond" data-replyto="Reply to orgesleka" aria-label="Reply to orgesleka">Reply</a> </div> </div> <div class="comment odd alt thread-even depth-1 vcard parent" id="comment-685201"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-685201" title="Permalink to this comment">23 June, 2024 at 1:04 pm</a></p> <p class="comment-author"><strong class="fn">Anonymous</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>The equation after the line “we conclude the trapezoidal rule” seems to have a truncated final O(N) expression in the latex.</p> <p><i>[I am not able to reproduce this issue either on my phone or my laptop. Can you clarify? -T.]</i></p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_685201"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=685201#respond" data-commentid="685201" data-postid="3643" data-belowelement="comment-685201" data-respondelement="respond" data-replyto="Reply to Anonymous" aria-label="Reply to Anonymous">Reply</a> </div> </div> <ul class="children"> <div class="comment even depth-2 vcard" id="comment-685211"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-685211" title="Permalink to this comment">24 June, 2024 at 6:36 pm</a></p> <p class="comment-author"><strong class="fn">Anonymous</strong></p> </div> <div class="comment-content"> <img alt='' src='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG' srcset='https://0.gravatar.com/avatar/?s=32&d=identicon&r=PG 1x, https://0.gravatar.com/avatar/?s=48&d=identicon&r=PG 1.5x, https://0.gravatar.com/avatar/?s=64&d=identicon&r=PG 2x, https://0.gravatar.com/avatar/?s=96&d=identicon&r=PG 3x, https://0.gravatar.com/avatar/?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Thanks for your quick reply. This is a screenshot from my company laptop. I tried zooming in/out or scrolling left/right but the closing ) was still missing. No worries if it’s too much work.</p> <figure class="wp-block-image size-medium"><img src="https://snipboard.io/gubJPB.jpg" alt="" /></figure> </p> <p><i>Strangely, I don’t get this overflow on my devices, but in any event I have broken up the display. -T]</i></p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_685211"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/?replytocom=685211#respond" data-commentid="685211" data-postid="3643" data-belowelement="comment-685211" data-respondelement="respond" data-replyto="Reply to Anonymous" aria-label="Reply to Anonymous">Reply</a> </div> </div> </ul><!-- .children --> <div class="comment odd alt thread-odd thread-alt depth-1 vcard" id="comment-685648"> <div class="comment-metadata"> <p class="comment-permalink"><a href="https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/comment-page-3/#comment-685648" title="Permalink to this comment">23 August, 2024 at 3:16 am</a></p> <p class="comment-author"><strong class="fn">Ankur Jain</strong></p> </div> <div class="comment-content"> <img alt='' src='https://2.gravatar.com/avatar/2608f31c006220d76744d57a5c00211925aeac221a4eaac148e2729db8f70d1c?s=32&d=identicon&r=PG' srcset='https://2.gravatar.com/avatar/2608f31c006220d76744d57a5c00211925aeac221a4eaac148e2729db8f70d1c?s=32&d=identicon&r=PG 1x, https://2.gravatar.com/avatar/2608f31c006220d76744d57a5c00211925aeac221a4eaac148e2729db8f70d1c?s=48&d=identicon&r=PG 1.5x, https://2.gravatar.com/avatar/2608f31c006220d76744d57a5c00211925aeac221a4eaac148e2729db8f70d1c?s=64&d=identicon&r=PG 2x, https://2.gravatar.com/avatar/2608f31c006220d76744d57a5c00211925aeac221a4eaac148e2729db8f70d1c?s=96&d=identicon&r=PG 3x, https://2.gravatar.com/avatar/2608f31c006220d76744d57a5c00211925aeac221a4eaac148e2729db8f70d1c?s=128&d=identicon&r=PG 4x' class='avatar avatar-32' height='32' width='32' loading='lazy' decoding='async' /> <p>Sir,<br />I want to bring this to your attention that i have been able to sum a divergent series to a finite sum using a Real Life Example of A Casino Game.</p> <p>Please find the PDF attached to this email.</p> <p>Any Feedback or Leads would be greatly appreciated.</p> </p> <p><a href="https://www.dropbox.com/scl/fi/i29gsjk4ku98nszt810g6/Gold-Nugget-2.pdf?rlkey=7j5stv317tgffdaqt36ll5367&st=1evr8vzb&dl=0" rel="nofollow ugc">https://www.dropbox.com/scl/fi/i29gsjk4ku98nszt810g6/Gold-Nugget-2.pdf?rlkey=7j5stv317tgffdaqt36ll5367&st=1evr8vzb&dl=0</a></p> <div class="cs-rating pd-rating" id="pd_rating_holder_132742_comm_685648"></div> </div> <div class="reply"> <a rel="nofollow" class="comment-reply-link" 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//--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_484116={"id":132742,"unique_id":"wp-comment-484116","title":"Greetings%20Prof.%20Tao%2COver%20the%20years%2CI%20have%20been%20working%20on%20the%20Riemann%20Hypothesis%20and%20I%20have%20recently%20found%20some%20results%20that%20might%20be%20interesting%20to%20you%20sir.I%20would%20like%20your%20comments%20and%20contrib...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-484116","item_id":"_comm_484116"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_485932={"id":132742,"unique_id":"wp-comment-485932","title":"%26%23091%3B%26%238230%3B%26%23093%3B%20you%20can%20read%20about%20divergent%20series%20in%20the%20following%20post%20by%20Terence%20Tao%20and%20in%20the%20book%20Divergent%20Series%20by%20Hardy.The%20Riemann%20hypothesis%20is%20an%20open%20problem%20in%20math%2C%20and%20%26%23091%3B%26%238230%3B%26%23093%3B...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-485932","item_id":"_comm_485932"}; 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//--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_505751={"id":132742,"unique_id":"wp-comment-505751","title":"First%20of%20all%2C%20you%20are%20wrong.%20The%20theorem%20does%20not%20contradict%20the%20basic%20principles%20of%20mathematics.%20In%20fact%2C%20I%20question%20whether%20you%20can%20even%20correctly%20list%20all%20the%20basic%20principles%20of%20mathematics%20y...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-505751","item_id":"_comm_505751"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_679774={"id":132742,"unique_id":"wp-comment-679774","title":"A%20theorem%20can%20imply%20a%20contradiction%20if%20the%20basics%20are%20inconsistent.%20Here%3A%20Only%20geometric%20series%20with%20q%20less%20than%201%20can%20be%20summed%20in%20a%20meaningful%20way.%20Further%20summing%20infinit%20sets%20obeys%20the%20same%20l...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-679774","item_id":"_comm_679774"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_511897={"id":132742,"unique_id":"wp-comment-511897","title":"%40RIvera%20I%20support%20your%20patience.%20A%20small%20linguistic%20remark.%20In%20the%20context%20of%20these%20%26%23039%3Bprovocative%26%23039%3B%20limits%20%28they%20are%20not%20a%20provocation%20at%20all%20to%20me.%29.%20I%20might%20be%20good%20avoid%20saying%20%26%23039%3Bthis%20series%20has%20...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-511897","item_id":"_comm_511897"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_519024={"id":132742,"unique_id":"wp-comment-519024","title":"May%20formula%20%2812%29%20not%20also%20be%20interpreted%20as%20a%20consequence%20or%20rather%20an%20instance%20of%20Ritt%26%23039%3Bs%20theorem%3F%20And%20therefore%20%2814%29%20a%20generalization%20thereof%3F%28Please%20excuse%20this%20naive%20question%20of%20a%20humble%20physi...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-519024","item_id":"_comm_519024"}; 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//--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_682669={"id":132742,"unique_id":"wp-comment-682669","title":"For%20functions%20of%20polynomial%20growth%20one%20can%20usually%20use%20the%20Euler-Maclaurin%20formula%20%2822%29%20to%20show%20that%20the%20cutoff%20%24latex%20eta%24%20only%20affects%20higher%20order%20terms%20and%20not%20the%20constant%20term.%20%20However%20fo...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-682669","item_id":"_comm_682669"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_682730={"id":132742,"unique_id":"wp-comment-682730","title":"You%20made%20a%20mistake.%20Equation%208%20is%20incorrect%2C%20you%20added%20each%20term%20in%20the%20series%20pointwise%2C%20which%20is%20a%20different%20operation%20than%20adding%202%20infinite%20series.%20Also%20consider%20pointwise%20multiplication%20of%20s...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-682730","item_id":"_comm_682730"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_682731={"id":132742,"unique_id":"wp-comment-682731","title":"Also%20sometimes%20he%20gives%203%20terms%20and%20in%20other%20cases%204.%20I%26%23039%3Bm%20sure%20there%26%23039%3Bs%20some%20significance%20to%20how%20many%20terms%20he%20gives.%20I%20think%20he%26%23039%3Bs%20giving%20the%20minimum%20number%20of%20terms%20needed%20to%20continue%20the%20pattern...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-682731","item_id":"_comm_682731"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_682732={"id":132742,"unique_id":"wp-comment-682732","title":"Also%20x2%20have%20you%20read%20the%20Aryabhatiyam%20by%20Aryabhata%3F%20I%26%23039%3Bm%20learning%20Sanskrit%20now%20using%20ai%20and%20I%20had%20it%20translate%20a%20bit%20and%20I%20read%20it%2C%20some%20parallels%20between%20it%20and%20ramunajan%26%23039%3Bs%20work.%20I%20can%26%23039%3Bt%20verify%20...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-682732","item_id":"_comm_682732"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_684439={"id":132742,"unique_id":"wp-comment-684439","title":"Related%3Ahttps%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3Dbeakj767uG4...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-684439","item_id":"_comm_684439"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_684452={"id":132742,"unique_id":"wp-comment-684452","title":"Is%20there%20a%20way%20of%20deriving%20the%20sum%20of%20a%20series%20from%20the%20sums%20of%20related%20series.%20For%20example%20is%20the%20sum%20of%20the%20natural%20numbers%20related%20in%20some%20way%20to%20the%20individual%20sums%20of%20the%20odd%20number%20and%20even...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-684452","item_id":"_comm_684452"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_684456={"id":132742,"unique_id":"wp-comment-684456","title":"There%20are%20some%20relations%2C%20but%20one%20needs%20to%20proceed%20with%20care%20as%20there%20are%20several%20pitfalls.%20%20Firstly%2C%20divergent%20summation%20methods%20tend%20to%20be%20linear%20in%20the%20summands%20%28provided%20that%20the%20sums%20actuall...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-684456","item_id":"_comm_684456"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_684459={"id":132742,"unique_id":"wp-comment-684459","title":"Thanks%20for%20replying%2C%20it%26%23039%3Bs%20fascinating.%20I%20came%20to%20you%20via%20your%20recent%20Numberphile%20appearance....","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-684459","item_id":"_comm_684459"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_684460={"id":132742,"unique_id":"wp-comment-684460","title":"Dear%20Terence%20Tao%2C%20I%20am%20not%20sure%20where%20to%20ask%20a%20question%20relating%20the%20Liouville%20function%20and%20an%20infinite%20dimensional%20lattice%2C%20so%20I%20thought%20to%20ask%20it%20here%3A%20Is%20this%20infinite%20dimensional%20lattice%20know...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-684460","item_id":"_comm_684460"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_685201={"id":132742,"unique_id":"wp-comment-685201","title":"%26lt%3B%21--%20wp%3Aparagraph%20--%26gt%3B%26lt%3Bp%26gt%3BThe%20equation%20after%20the%20line%20%26quot%3Bwe%20conclude%20the%20trapezoidal%20rule%26quot%3B%20seems%20to%20have%20a%20truncated%20final%20O%28N%29%20expression%20in%20the%20latex.%26lt%3B%2Fp%26gt%3B%26lt%3B%21--%20%2Fwp%3Aparagraph%20--%26gt%3B%26lt%3Bi%26gt%3B%26%23091%3BI%20am%20not%20able%20to%20...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-685201","item_id":"_comm_685201"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_685211={"id":132742,"unique_id":"wp-comment-685211","title":"%26lt%3B%21--%20wp%3Aparagraph%20--%26gt%3B%26lt%3Bp%26gt%3BThanks%20for%20your%20quick%20reply.%20This%20is%20a%20screenshot%20from%20my%20company%20laptop.%20I%20tried%20zooming%20in%2Fout%20or%20scrolling%20left%2Fright%20but%20%20the%20closing%20%29%20was%20still%20missing.%20No%20worries%20i...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-685211","item_id":"_comm_685211"}; //--><!]]]]><![CDATA[> <!--//--><![CDATA[//><!-- PDRTJS_settings_132742_comm_685648={"id":132742,"unique_id":"wp-comment-685648","title":"%26lt%3B%21--%20wp%3Aparagraph%20--%26gt%3B%26lt%3Bp%26gt%3BSir%2C%26lt%3Bbr%20%2F%26gt%3BI%20want%20to%20bring%20this%20to%20your%20attention%20that%20i%20have%20been%20able%20to%20sum%20a%20divergent%20series%20to%20a%20finite%20sum%20using%20a%20Real%20Life%20Example%20of%20A%20Casino%20Game.%26lt%3B%2Fp%26gt%3B%26lt%3B%21--%20%2Fwp%3Apa...","permalink":"https:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/#comment-685648","item_id":"_comm_685648"}; 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