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<!doctype html> <html lang="en" class="calculusandanalysis foundationsofmathematics interactiveentries mathworldcontributors"> <head> <title>Riemann Zeta Function -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Riemann Zeta Function" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day. 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History and Terminology </a> <a href="/topics/NumberTheory.html" id="sidebar-numbertheory"> Number Theory </a> <a href="/topics/ProbabilityandStatistics.html" id="sidebar-probabilityandstatistics"> Probability and Statistics </a> <a href="/topics/RecreationalMathematics.html" id="sidebar-recreationalmathematics"> Recreational Mathematics </a> <a href="/topics/Topology.html" id="sidebar-topology"> Topology </a> </nav> <nav class="secondary-nav"> <a href="/letters/"> Alphabetical Index </a> <a href="/whatsnew/"> New in MathWorld </a> </nav> </section> <section id="content">  <nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/CalculusandAnalysis.html">Calculus and Analysis</a> </li> <li> <a href="/topics/SpecialFunctions.html">Special Functions</a> </li> <li> <a href="/topics/RiemannZetaFunction.html">Riemann Zeta Function</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/FoundationsofMathematics.html">Foundations of Mathematics</a> </li> <li> <a 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</ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Sondow.html">Sondow</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Terr.html">Terr</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Trott.html">Trott</a> </li> </ul><a class="show-more">More...</a><a class="display-n show-less">Less...</a></nav>   <h1>Riemann Zeta Function</h1>  <hr class="margin-t-1-8 margin-b-3-4">  <div class="attachments text-align-r"> <a href="/notebooks/SpecialFunctions/RiemannZetaFunction.nb" download="RiemannZetaFunction.nb"><img src="/images/entries/download-notebook-icon.png" width="26" height="27" alt="DOWNLOAD Mathematica Notebook" /><span>Download <span class="display-i display-n__600">Wolfram </span>Notebook</span></a> </div>  <div class="entry-content"> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="555.86" height="171.522" src="images/eps-svg/RiemannZeta_1002.svg" class="" alt="RiemannZeta" /> </div> <p> The Riemann zeta function is an extremely important <a href="/SpecialFunction.html">special function</a> of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the <a href="/PrimeNumberTheorem.html">prime number theorem</a>. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the <a href="/RiemannHypothesis.html">Riemann hypothesis</a>) that remain unproved to this day. The Riemann zeta function is denoted <img src="/images/equations/RiemannZetaFunction/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="zeta(s)" /> and is plotted above (using two different scales) along the real axis. </p> <table width="100%" summary="" title="" align="center"> <tr> <td align="center"><img alt="RiemannZetaReImAbs" name="RiemannZetaReImAbs" src="/images/interactive/RiemannZetaReImAbs.gif" id="RiemannZetaReImAbs" /></td> </tr> <tr> <td align="center"> <form name="RiemannZetaReImAbsoptions" method="post" action="#" id="RiemannZetaReImAbsoptions"> <table width="450" summary="" cellspacing="0" cellpadding="0" border="0" bgcolor="#F9FFED"> <tr> <td bgcolor="#9BCDB9"></td> </tr> <tr> <td height="1" bgcolor="#9BCDB9"><img width="1" height="1" alt="" src="/images/spacer.gif" /></td> </tr> <tr> <td height="1" bgcolor="#EDF6B8"><img width="1" height="1" alt="" src="/images/spacer.gif" /></td> </tr> <tr> <td height="1" bgcolor="#9BCDB9"><img width="1" height="1" alt="" src="/images/spacer.gif" /></td> </tr> <tr> <td bgcolor="#9BCDB9"></td> </tr> <tr> <td bgcolor="#F8FFEC"> <table summary="" cellspacing="0" cellpadding="2" border="0" align="center"> <tr> <td><img width="4" height="1" alt="" src="/images/spacer.gif" /></td> <td></td> <td>Min</td> <td></td> <td>Max</td> <td></td> <td></td> <td><img width="4" height="1" alt="" src="/images/spacer.gif" /></td> </tr> <tr> <td></td> <td>Re</td> <td><input type="text" value="-15" size="6" name="ReMin" /></td> <td></td> <td><input type="text" value="15" size="6" name="ReMax" /></td> <td></td> </tr> <tr> <td></td> <td>Im</td> <td><input type="text" value="-15" size="6" name="ImMin" /></td> <td></td> <td><input type="text" value="15" size="6" name="ImMax" /></td> <td><input type="button" name="plotbutton" value="Replot" onclick="document.images.RiemannZetaReImAbs.src=showComplexPlots("RiemannZeta",document.forms.RiemannZetaReImAbsoptions.ReMin.value,document.forms.RiemannZetaReImAbsoptions.ReMax.value,document.forms.RiemannZetaReImAbsoptions.ImMin.value,document.forms.RiemannZetaReImAbsoptions.ImMax.value);return false;" /></td> <td align="center"><a target="new" href="https://www.wolfram.com/products/webmathematica/"><img width="142" height="28" border="0" alt="Powered by webMathematica" src="/images/entries/webM-mw3.gif" /></a></td> <td></td> </tr> </table> </td> </tr> </table> </form> </td> </tr> </table> <p> In general, <img src="/images/equations/RiemannZetaFunction/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="zeta(s)" /> is defined over the complex plane for one complex variable, which is conventionally denoted <img src="/images/equations/RiemannZetaFunction/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="7" height="21" alt="s" /> (instead of the usual <img src="/images/equations/RiemannZetaFunction/Inline4.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="z" />) in deference to the notation used by Riemann in his 1859 paper that founded the study of this function (Riemann 1859). </p> <p> <img src="/images/equations/RiemannZetaFunction/Inline5.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="zeta(s)" /> is implemented in the <a href="http://www.wolfram.com/language/">Wolfram Language</a> as <tt><a href="http://reference.wolfram.com/language/ref/Zeta.html">Zeta</a></tt>[<i>s</i>]. </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="255.607" height="207.266" src="images/eps-svg/RiemannZetaRidges_700.svg" class="" alt="RiemannZetaRidges" /> </div> <p> The plot above shows the "ridges" of <img src="/images/equations/RiemannZetaFunction/Inline6.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="76" height="21" alt="|zeta(x+iy)|" /> for <img src="/images/equations/RiemannZetaFunction/Inline7.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="65" height="21" alt="0<x<1" /> and <img src="/images/equations/RiemannZetaFunction/Inline8.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="83" height="21" alt="1<y<100" />. The fact that the ridges appear to decrease monotonically for <img src="/images/equations/RiemannZetaFunction/Inline9.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="85" height="21" alt="0<=x<=1/2" /> is not a coincidence since it turns out that monotonic decrease implies the <a href="/RiemannHypothesis.html">Riemann hypothesis</a> (Zvengrowski and Saidak 2003; Borwein and Bailey 2003, pp. 95-96). </p> <p> On the <a href="/RealLine.html">real line</a> with <img src="/images/equations/RiemannZetaFunction/Inline10.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="x>1" />, the Riemann zeta function can be defined by the integral </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation1.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="186" height="45" alt=" zeta(x)=1/(Gamma(x))int_0^infty(u^(x-1))/(e^u-1)du, " /></td><td align="right" width="3"> <div id="eqn1" class="eqnum"> (1) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline11.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="33" height="21" alt="Gamma(x)" /> is the <a href="/GammaFunction.html">gamma function</a>. If <img src="/images/equations/RiemannZetaFunction/Inline12.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="x" /> is an <a href="/Integer.html">integer</a> <img src="/images/equations/RiemannZetaFunction/Inline13.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />, then we have the identity </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline14.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="46" height="43" alt="(u^(n-1))/(e^u-1)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline15.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline16.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="60" height="43" alt="(e^(-u)u^(n-1))/(1-e^(-u))" /></td><td align="right" width="10"> <div id="eqn2" class="eqnum"> (2) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline17.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline18.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline19.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="111" height="50" alt="e^(-u)u^(n-1)sum_(k=0)^(infty)e^(-ku)" /></td><td align="right" width="10"> <div id="eqn3" class="eqnum"> (3) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline20.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline21.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline22.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="90" height="50" alt="sum_(k=1)^(infty)e^(-ku)u^(n-1)," /></td><td align="right" width="10"> <div id="eqn4" class="eqnum"> (4) </div> </td></tr> </table> </div> <p> so </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation2.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="257" height="51" alt=" int_0^infty(u^(n-1))/(e^u-1)du=sum_(k=1)^inftyint_0^inftye^(-ku)u^(n-1)du. " /></td><td align="right" width="3"> <div id="eqn5" class="eqnum"> (5) </div> </td></tr> </table> </div> <p> To evaluate <img src="/images/equations/RiemannZetaFunction/Inline23.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(n)" />, let <img src="/images/equations/RiemannZetaFunction/Inline24.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="50" height="21" alt="y=ku" /> so that <img src="/images/equations/RiemannZetaFunction/Inline25.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="80" height="21" alt="dy=kdu" /> and plug in the above identity to obtain </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline26.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(n)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline27.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline28.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="176" height="50" alt="1/(Gamma(n))sum_(k=1)^(infty)int_0^inftye^(-ku)u^(n-1)du" /></td><td align="right" width="10"> <div id="eqn6" class="eqnum"> (6) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline29.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline30.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline31.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="192" height="50" alt="1/(Gamma(n))sum_(k=1)^(infty)int_0^inftye^(-y)(y/k)^(n-1)(dy)/k" /></td><td align="right" width="10"> <div id="eqn7" class="eqnum"> (7) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline32.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline33.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline34.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="196" height="50" alt="1/(Gamma(n))sum_(k=1)^(infty)1/(k^n)int_0^inftye^(-y)y^(n-1)dy." /></td><td align="right" width="10"> <div id="eqn8" class="eqnum"> (8) </div> </td></tr> </table> </div> <p> Integrating the final expression in (<a href="#eqn8">8</a>) gives <img src="/images/equations/RiemannZetaFunction/Inline35.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="33" height="21" alt="Gamma(n)" />, which cancels the factor <img src="/images/equations/RiemannZetaFunction/Inline36.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="53" height="21" alt="1/Gamma(n)" /> and gives the most common form of the Riemann zeta function, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation3.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="100" height="50" alt=" zeta(n)=sum_(k=1)^infty1/(k^n), " /></td><td align="right" width="3"> <div id="eqn9" class="eqnum"> (9) </div> </td></tr> </table> </div> <p> which is sometimes known as a <a href="/p-Series.html"><i>p</i>-series</a>. </p> <p> The Riemann zeta function can also be defined in terms of <a href="/MultipleIntegral.html">multiple integrals</a> by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation4.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="200" height="61" alt=" zeta(n)=int_0^1...int_0^1_()_(n)(product_(i=1)^(n)dx_i)/(1-product_(i=1)^(n)x_i), " /></td><td align="right" width="3"> <div id="eqn10" class="eqnum"> (10) </div> </td></tr> </table> </div> <p> and as a <a href="/MellinTransform.html">Mellin transform</a> by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation5.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="195" height="41" alt=" int_0^inftyfrac(1/t)t^(s-1)dt=-(zeta(s))/s " /></td><td align="right" width="3"> <div id="eqn11" class="eqnum"> (11) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/RiemannZetaFunction/Inline37.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="91" height="21" alt="0<R[s]<1" />, where <img src="/images/equations/RiemannZetaFunction/Inline38.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="51" height="21" alt="frac(x)" /> is the <a href="/FractionalPart.html">fractional part</a> (Balazard and Saias 2000). </p> <p> It appears in the <a href="/UnitSquareIntegral.html">unit square integral</a> </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation6.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="307" height="44" alt=" int_0^1int_0^1([-ln(xy)]^s)/(1-xy)dxdy=Gamma(s+2)zeta(s+2), " /></td><td align="right" width="3"> <div id="eqn12" class="eqnum"> (12) </div> </td></tr> </table> </div> <p> valid for <img src="/images/equations/RiemannZetaFunction/Inline39.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="63" height="21" alt="R[s]>1" /> (Guillera and Sondow 2005). For <img src="/images/equations/RiemannZetaFunction/Inline40.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="7" height="21" alt="s" /> a nonnegative integer, this formula is due to Hadjicostas (2002), and the special cases <img src="/images/equations/RiemannZetaFunction/Inline41.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="s=0" /> and <img src="/images/equations/RiemannZetaFunction/Inline42.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="s=1" /> are due to Beukers (1979). </p> <p> Note that the zeta function <img src="/images/equations/RiemannZetaFunction/Inline43.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="zeta(s)" /> has a singularity at <img src="/images/equations/RiemannZetaFunction/Inline44.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="s=1" />, where it reduces to the divergent <a href="/HarmonicSeries.html">harmonic series</a>. </p> <p> The Riemann zeta function satisfies the <a href="/ReflectionRelation.html">reflection</a> <a href="/FunctionalEquation.html">functional equation</a> </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation7.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="264" height="26" alt=" zeta(1-s)=2(2pi)^(-s)cos(1/2spi)Gamma(s)zeta(s) " /></td><td align="right" width="3"> <div id="eqn13" class="eqnum"> (13) </div> </td></tr> </table> </div> <p> (Hardy 1999, p. 14; Krantz 1999, p. 160), a similar form of which was conjectured by Euler for real <img src="/images/equations/RiemannZetaFunction/Inline45.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="7" height="21" alt="s" /> (Euler, read in 1749, published in 1768; Ayoub 1974; Havil 2003, p. 193). A symmetrical form of this functional equation is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation8.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="296" height="40" alt=" Gamma(s/2)pi^(-s/2)zeta(s)=Gamma((1-s)/2)pi^(-(1-s)/2)zeta(1-s) " /></td><td align="right" width="3"> <div id="eqn14" class="eqnum"> (14) </div> </td></tr> </table> </div> <p> (Ayoub 1974), which was proved by Riemann for all complex <img src="/images/equations/RiemannZetaFunction/Inline46.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="7" height="21" alt="s" /> (Riemann 1859). </p> <p> As defined above, the zeta function <img src="/images/equations/RiemannZetaFunction/Inline47.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="zeta(s)" /> with <img src="/images/equations/RiemannZetaFunction/Inline48.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="73" height="21" alt="s=sigma+it" /> a <a href="/ComplexNumber.html">complex number</a> is defined for <img src="/images/equations/RiemannZetaFunction/Inline49.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="63" height="21" alt="R[s]>1" />. However, <img src="/images/equations/RiemannZetaFunction/Inline50.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="zeta(s)" /> has a unique <a href="/AnalyticContinuation.html">analytic continuation</a> to the entire <a href="/ComplexPlane.html">complex plane</a>, excluding the point <img src="/images/equations/RiemannZetaFunction/Inline51.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="s=1" />, which corresponds to a <a href="/SimplePole.html">simple pole</a> with <a href="/ComplexResidue.html">complex residue</a> 1 (Krantz 1999, p. 160). In particular, as <img src="/images/equations/RiemannZetaFunction/Inline52.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="21" alt="s->1" />, <img src="/images/equations/RiemannZetaFunction/Inline53.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="zeta(s)" /> obeys </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation9.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="153" height="40" alt=" lim_(s->1)[zeta(s)-1/(s-1)]=gamma, " /></td><td align="right" width="3"> <div id="eqn15" class="eqnum"> (15) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline54.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="gamma" /> is the <a href="/Euler-MascheroniConstant.html">Euler-Mascheroni constant</a> (Whittaker and Watson 1990, p. 271). </p> <p> To perform the <a href="/AnalyticContinuation.html">analytic continuation</a> for <img src="/images/equations/RiemannZetaFunction/Inline55.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="63" height="21" alt="R[s]>0" />, write </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline56.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="128" height="49" alt="sum_(n=1)^(infty)((-1)^n)/(n^s)+sum_(n=1)^(infty)1/(n^s)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline57.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline58.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="80" height="51" alt="2sum_(n=2,4,...)^(infty)1/(n^s)" /></td><td align="right" width="10"> <div id="eqn16" class="eqnum"> (16) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline59.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline60.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline61.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="78" height="50" alt="2sum_(k=1)^(infty)1/((2k)^s)" /></td><td align="right" width="10"> <div id="eqn17" class="eqnum"> (17) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline62.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline63.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline64.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="79" height="50" alt="2^(1-s)sum_(k=1)^(infty)1/(k^s)," /></td><td align="right" width="10"> <div id="eqn18" class="eqnum"> (18) </div> </td></tr> </table> </div> <p> so rewriting in terms of <img src="/images/equations/RiemannZetaFunction/Inline65.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="zeta(s)" /> immediately gives </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation10.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="195" height="49" alt=" sum_(n=1)^infty((-1)^n)/(n^s)+zeta(s)=2^(1-s)zeta(s). " /></td><td align="right" width="3"> <div id="eqn19" class="eqnum"> (19) </div> </td></tr> </table> </div> <p> Therefore, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation11.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="195" height="50" alt=" zeta(s)=1/(1-2^(1-s))sum_(n=1)^infty((-1)^(n-1))/(n^s). " /></td><td align="right" width="3"> <div id="eqn20" class="eqnum"> (20) </div> </td></tr> </table> </div> <p> Here, the sum on the right-hand side is exactly the <a href="/DirichletEtaFunction.html">Dirichlet eta function</a> <img src="/images/equations/RiemannZetaFunction/Inline66.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="eta(s)" /> (sometimes also called the alternating zeta function). While this formula defines <img src="/images/equations/RiemannZetaFunction/Inline67.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="zeta(s)" /> for only the <a href="/RightHalf-Plane.html">right half-plane</a> <img src="/images/equations/RiemannZetaFunction/Inline68.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="63" height="21" alt="R[s]>0" />, equation (◇) can be used to analytically continue it to the rest of the <a href="/ComplexPlane.html">complex plane</a>. <a href="/AnalyticContinuation.html">Analytic continuation</a> can also be performed using <a href="/HankelFunction.html">Hankel functions</a>. A globally convergent series for the Riemann zeta function (which provides the <a href="/AnalyticContinuation.html">analytic continuation</a> of <img src="/images/equations/RiemannZetaFunction/Inline69.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="zeta(s)" /> to the entire <a href="/ComplexPlane.html">complex plane</a> except <img src="/images/equations/RiemannZetaFunction/Inline70.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="s=1" />) is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation12.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="323" height="51" alt=" zeta(s)=1/(1-2^(1-s))sum_(n=0)^infty1/(2^(n+1))sum_(k=0)^n(-1)^k(n; k)(k+1)^(-s) " /></td><td align="right" width="3"> <div id="eqn21" class="eqnum"> (21) </div> </td></tr> </table> </div> <p> (Havil 2003, p. 206), where <img src="/images/equations/RiemannZetaFunction/Inline71.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="43" alt="(n; k)" /> is a <a href="/BinomialCoefficient.html">binomial coefficient</a>, which was conjectured by Knopp around 1930, proved by Hasse (1930), and rediscovered by Sondow (1994). This equation is related to renormalization and random variates (Biane <i>et al. </i>2001) and can be derived by applying <a href="/EulersSeriesTransformation.html">Euler's series transformation</a> with <img src="/images/equations/RiemannZetaFunction/Inline72.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="n=0" /> to equation (<a href="#eqn20">20</a>). </p> <p> Hasse (1930) also proved the related globally (but more slowly) convergent series </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation13.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="313" height="51" alt=" zeta(s)=1/(s-1)sum_(n=0)^infty1/(n+1)sum_(k=0)^n(-1)^k(n; k)(k+1)^(1-s) " /></td><td align="right" width="3"> <div id="eqn22" class="eqnum"> (22) </div> </td></tr> </table> </div> <p> that, unlike (<a href="#eqn21">21</a>), can also be extended to a generalization of the Riemann zeta function known as the <a href="/HurwitzZetaFunction.html">Hurwitz zeta function</a> <img src="/images/equations/RiemannZetaFunction/Inline73.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="48" height="21" alt="zeta(s,a)" />. <img src="/images/equations/RiemannZetaFunction/Inline74.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="48" height="21" alt="zeta(s,a)" /> is defined such that </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation14.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="96" height="20" alt=" zeta(s)=zeta(s,1). " /></td><td align="right" width="3"> <div id="eqn23" class="eqnum"> (23) </div> </td></tr> </table> </div> <p> (If the singular term is excluded from the sum definition of <img src="/images/equations/RiemannZetaFunction/Inline75.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="48" height="21" alt="zeta(s,a)" />, then <img src="/images/equations/RiemannZetaFunction/Inline76.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="98" height="21" alt="zeta(s)=zeta(s,0)" /> as well.) Expanding <img src="/images/equations/RiemannZetaFunction/Inline77.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="zeta(s)" /> about <img src="/images/equations/RiemannZetaFunction/Inline78.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="s=1" /> gives </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation15.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="244" height="49" alt=" zeta(s)=1/(s-1)+sum_(n=0)^infty((-1)^n)/(n!)gamma_n(s-1)^n, " /></td><td align="right" width="3"> <div id="eqn24" class="eqnum"> (24) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline79.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="22" alt="gamma_n" /> are the so-called <a href="/StieltjesConstants.html">Stieltjes constants</a>. </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="223.263" height="213.864" src="images/eps-svg/RiemannZetaFunctionGamma_1000.svg" class="" alt="RiemannZetaFunctionGamma" /> </div> <p> The Riemann zeta function can also be defined in the complex plane by the <a href="/ContourIntegral.html">contour integral</a> </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation16.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="205" height="46" alt=" zeta(z)=(Gamma(1-z))/(2pii)∮_gamma(u^(z-1))/(e^(-u)-1)du " /></td><td align="right" width="3"> <div id="eqn25" class="eqnum"> (25) </div> </td></tr> </table> </div> <p> for all <img src="/images/equations/RiemannZetaFunction/Inline80.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="36" height="21" alt="z!=1" />, where the <a href="/Contour.html">contour</a> is illustrated above (Havil 2003, pp. 193 and 249-252). </p> <p> Zeros of <img src="/images/equations/RiemannZetaFunction/Inline81.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="zeta(s)" /> come in (at least) two different types. So-called "trivial zeros" occur at <i>all</i> negative even integers <img src="/images/equations/RiemannZetaFunction/Inline82.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="49" height="21" alt="s=-2" />, <img src="/images/equations/RiemannZetaFunction/Inline83.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="21" height="21" alt="-4" />, <img src="/images/equations/RiemannZetaFunction/Inline84.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="21" height="21" alt="-6" />, ..., and "nontrivial zeros" at certain </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation17.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="65" height="20" alt=" s=sigma+it " /></td><td align="right" width="3"> <div id="eqn26" class="eqnum"> (26) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/RiemannZetaFunction/Inline85.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="7" height="21" alt="s" /> in the "<a href="/CriticalStrip.html">critical strip</a>" <img src="/images/equations/RiemannZetaFunction/Inline86.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="68" height="21" alt="0<sigma<1" />. The <a href="/RiemannHypothesis.html">Riemann hypothesis</a> asserts that the nontrivial <a href="/RiemannZetaFunctionZeros.html">Riemann zeta function zeros</a> of <img src="/images/equations/RiemannZetaFunction/Inline87.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="zeta(s)" /> all have <a href="/RealPart.html">real part</a> <img src="/images/equations/RiemannZetaFunction/Inline88.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="118" height="21" alt="sigma=R[s]=1/2" />, a line called the "<a href="/CriticalLine.html">critical line</a>." This is now known to be true for the first <img src="/images/equations/RiemannZetaFunction/Inline89.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="69" height="21" alt="250×10^9" /> roots. </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="255.607" height="207.41" src="images/eps-svg/RiemannZetaCriticalStrip_700.svg" class="" alt="RiemannZetaCriticalStrip" /> </div> <p> The plot above shows the real and imaginary parts of <img src="/images/equations/RiemannZetaFunction/Inline90.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="88" height="21" alt="zeta(1/2+iy)" /> (i.e., values of <img src="/images/equations/RiemannZetaFunction/Inline91.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="30" height="21" alt="zeta(z)" /> along the <a href="/CriticalLine.html">critical line</a>) as <img src="/images/equations/RiemannZetaFunction/Inline92.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="y" /> is varied from 0 to 35 (Derbyshire 2004, p. 221). </p> <p> The Riemann zeta function can be split up into </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation18.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="158" height="26" alt=" zeta(1/2+it)=Z(t)e^(-itheta(t)), " /></td><td align="right" width="3"> <div id="eqn27" class="eqnum"> (27) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline93.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="30" height="21" alt="Z(t)" /> and <img src="/images/equations/RiemannZetaFunction/Inline94.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="27" height="21" alt="theta(t)" /> are the <a href="/Riemann-SiegelFunctions.html">Riemann-Siegel functions</a>. </p> <p> The Riemann zeta function is related to the <a href="/DirichletLambdaFunction.html">Dirichlet lambda function</a> <img src="/images/equations/RiemannZetaFunction/Inline95.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="lambda(nu)" /> and <a href="/DirichletEtaFunction.html">Dirichlet eta function</a> <img src="/images/equations/RiemannZetaFunction/Inline96.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="30" height="21" alt="eta(nu)" /> by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation19.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="168" height="38" alt=" (zeta(nu))/(2^nu)=(lambda(nu))/(2^nu-1)=(eta(nu))/(2^nu-2) " /></td><td align="right" width="3"> <div id="eqn28" class="eqnum"> (28) </div> </td></tr> </table> </div> <p> and </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation20.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="133" height="20" alt=" zeta(nu)+eta(nu)=2lambda(nu) " /></td><td align="right" width="3"> <div id="eqn29" class="eqnum"> (29) </div> </td></tr> </table> </div> <p> (Spanier and Oldham 1987). </p> <p> It is related to the <a href="/LiouvilleFunction.html">Liouville function</a> <img src="/images/equations/RiemannZetaFunction/Inline97.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="21" alt="lambda(n)" /> by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation21.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="127" height="49" alt=" (zeta(2s))/(zeta(s))=sum_(n=1)^infty(lambda(n))/(n^s) " /></td><td align="right" width="3"> <div id="eqn30" class="eqnum"> (30) </div> </td></tr> </table> </div> <p> (Lehman 1960, Hardy and Wright 1979). Furthermore, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation22.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="135" height="50" alt=" (zeta^2(s))/(zeta(2s))=sum_(n=1)^infty(2^(omega(n)))/(n^s), " /></td><td align="right" width="3"> <div id="eqn31" class="eqnum"> (31) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline98.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="35" height="21" alt="omega(n)" /> is the number of <a href="/DistinctPrimeFactors.html">distinct prime factors</a> of <img src="/images/equations/RiemannZetaFunction/Inline99.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> (Hardy and Wright 1979, p. 254). </p> <p> For <img src="/images/equations/RiemannZetaFunction/Inline100.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="34" height="21" alt="-2n" /> a positive even integer <img src="/images/equations/RiemannZetaFunction/Inline101.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="21" height="21" alt="-2" />, <img src="/images/equations/RiemannZetaFunction/Inline102.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="21" height="21" alt="-4" />, ..., </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation23.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="237" height="42" alt=" zeta^'(-2n)=((-1)^nzeta(2n+1)(2n)!)/(2^(2n+1)pi^(2n)), " /></td><td align="right" width="3"> <div id="eqn32" class="eqnum"> (32) </div> </td></tr> </table> </div> <p> giving the first few as </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline103.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="46" height="21" alt="zeta^'(-2)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline104.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline105.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="47" height="42" alt="-(zeta(3))/(4pi^2)" /></td><td align="right" width="10"> <div id="eqn33" class="eqnum"> (33) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline106.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="46" height="21" alt="zeta^'(-4)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline107.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline108.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="48" height="42" alt="(3zeta(5))/(4pi^4)" /></td><td align="right" width="10"> <div id="eqn34" class="eqnum"> (34) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline109.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="46" height="21" alt="zeta^'(-6)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline110.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline111.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="67" height="42" alt="-(45zeta(7))/(8pi^6)" /></td><td align="right" width="10"> <div id="eqn35" class="eqnum"> (35) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline112.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="46" height="21" alt="zeta^'(-8)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline113.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline114.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="65" height="42" alt="(315zeta(9))/(4pi^8)" /></td><td align="right" width="10"> <div id="eqn36" class="eqnum"> (36) </div> </td></tr> </table> </div> <p> (OEIS <a href="http://oeis.org/A117972">A117972</a> and <a href="http://oeis.org/A117973">A117973</a>). For <img src="/images/equations/RiemannZetaFunction/Inline115.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="51" height="21" alt="n=-1" />, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation24.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="132" height="26" alt=" zeta^'(-1)=1/(12)-lnA, " /></td><td align="right" width="3"> <div id="eqn37" class="eqnum"> (37) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline116.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="A" /> is the <a href="/Glaisher-KinkelinConstant.html">Glaisher-Kinkelin constant</a>. Using equation (◇) gives the derivative </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation25.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="132" height="26" alt=" zeta^'(0)=-1/2ln(2pi), " /></td><td align="right" width="3"> <div id="eqn38" class="eqnum"> (38) </div> </td></tr> </table> </div> <p> which can be derived directly from the <a href="/WallisFormula.html">Wallis formula</a> (Sondow 1994). <img src="/images/equations/RiemannZetaFunction/Inline117.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="149" height="22" alt="zeta^'(0)/zeta(0)=ln(2pi)" /> can also be derived directly from the Euler-Maclaurin summation formula (Edwards 2001, pp. 134-135). In general, <img src="/images/equations/RiemannZetaFunction/Inline118.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="46" height="22" alt="zeta^((n))(0)" /> can be expressed analytically in terms of <img src="/images/equations/RiemannZetaFunction/Inline119.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" />, <img src="/images/equations/RiemannZetaFunction/Inline120.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(n)" />, the <a href="/Euler-MascheroniConstant.html">Euler-Mascheroni constant</a> <img src="/images/equations/RiemannZetaFunction/Inline121.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="gamma" />, and the <a href="/StieltjesConstants.html">Stieltjes constants</a> <img src="/images/equations/RiemannZetaFunction/Inline122.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="14" height="22" alt="gamma_i" />, with the first few examples being </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline123.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="zeta^('')(0)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline124.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline125.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="207" height="26" alt="gamma_1+1/2gamma^2-1/(24)pi^2-1/2[ln(2pi)]^2" /></td><td align="right" width="10"> <div id="eqn39" class="eqnum"> (39) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline126.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="42" height="21" alt="zeta^(''')(0)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline127.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline128.svg" data-src-small="/images/equations/RiemannZetaFunction/Inline128_400.svg" data-src-default="/images/equations/RiemannZetaFunction/Inline128.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="539" height="26" data-big="539 26" data-small="317 56" border="0" alt="3ln(2pi)gamma_1+3gammagamma_1+3/2gamma_2-zeta(3)-1/2[ln(2pi)]^3-1/8pi^2ln(2pi)+3/2gamma^2ln(2pi)+gamma^3." /></td><td align="right" width="10"> <div id="eqn40" class="eqnum"> (40) </div> </td></tr> </table> </div> <p> Derivatives <img src="/images/equations/RiemannZetaFunction/Inline129.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="66" height="22" alt="zeta^((n))(1/2)" /> can also be given in closed form, for example, </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline130.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="38" height="26" alt="zeta^'(1/2)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline131.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline132.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="220" height="26" alt="1/4[(pi+2gamma+6ln2+2lnpi)zeta(1/2)]" /></td><td align="right" width="10"> <div id="eqn41" class="eqnum"> (41) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline133.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline134.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline135.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="111" height="20" alt="-3.92264613..." /></td><td align="right" width="10"> <div id="eqn42" class="eqnum"> (42) </div> </td></tr> </table> </div> <p> (OEIS <a href="http://oeis.org/A114875">A114875</a>). </p> <p> The <a href="/Derivative.html">derivative</a> of the Riemann zeta function for <img src="/images/equations/RiemannZetaFunction/Inline136.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="63" height="21" alt="R[s]>1" /> is defined by </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline137.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="33" height="21" alt="zeta^'(s)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline138.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline139.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="65" height="50" alt="-sum_(k=1)^(infty)(lnk)/(k^s)" /></td><td align="right" width="10"> <div id="eqn43" class="eqnum"> (43) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline140.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline141.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline142.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="69" height="50" alt="-sum_(k=2)^(infty)(lnk)/(k^s)." /></td><td align="right" width="10"> <div id="eqn44" class="eqnum"> (44) </div> </td></tr> </table> </div> <p> <img src="/images/equations/RiemannZetaFunction/Inline143.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="36" height="22" alt="zeta^'(2)" /> can be given in closed form as </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline144.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="35" height="21" alt="zeta^'(2)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline145.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline146.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="176" height="26" alt="1/6pi^2[gamma+ln(2pi)-12lnA]" /></td><td align="right" width="10"> <div id="eqn45" class="eqnum"> (45) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline147.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline148.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline149.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="136" height="20" alt="-0.93754825431..." /></td><td align="right" width="10"> <div id="eqn46" class="eqnum"> (46) </div> </td></tr> </table> </div> <p> (OEIS <a href="http://oeis.org/A073002">A073002</a>), where <img src="/images/equations/RiemannZetaFunction/Inline150.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="A" /> is the <a href="/Glaisher-KinkelinConstant.html">Glaisher-Kinkelin constant</a> (given in series form by Glaisher 1894). </p> <p> The series for <img src="/images/equations/RiemannZetaFunction/Inline151.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="34" height="22" alt="zeta^'(s)" /> about <img src="/images/equations/RiemannZetaFunction/Inline152.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="s=1" /> is </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation26.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="361" height="44" alt=" zeta^'(s)=-1/((s-1)^2)-gamma_1+gamma_2(s-1)-1/2gamma_3(s-1)^2+..., " /></td><td align="right" width="3"> <div id="eqn47" class="eqnum"> (47) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline153.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="14" height="22" alt="gamma_i" /> are <a href="/StieltjesConstants.html">Stieltjes constants</a>. </p> <p> In 1739, Euler found the rational coefficients <img src="/images/equations/RiemannZetaFunction/Inline154.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="C" /> in <img src="/images/equations/RiemannZetaFunction/Inline155.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="107" height="21" alt="zeta(2n)=Cpi^(2n)" /> in terms of the <a href="/BernoulliNumber.html">Bernoulli numbers</a>. Which, when combined with the 1882 proof by Lindemann that <img src="/images/equations/RiemannZetaFunction/Inline156.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> is transcendental, effectively proves that <img src="/images/equations/RiemannZetaFunction/Inline157.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="44" height="21" alt="zeta(2n)" /> is transcendental. The study of <img src="/images/equations/RiemannZetaFunction/Inline158.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="72" height="21" alt="zeta(2n+1)" /> is significantly more difficult. Apéry (1979) finally proved <img src="/images/equations/RiemannZetaFunction/Inline159.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(3)" /> to be <a href="/IrrationalNumber.html">irrational</a>, but no similar results are known for other <a href="/OddNumber.html">odd</a> <img src="/images/equations/RiemannZetaFunction/Inline160.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />. As a result of Apéry's important discovery, <img src="/images/equations/RiemannZetaFunction/Inline161.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(3)" /> is sometimes called <a href="/AperysConstant.html">Apéry's constant</a>. Rivoal (2000) and Ball and Rivoal (2001) proved that there are infinitely many integers <img src="/images/equations/RiemannZetaFunction/Inline162.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> such that <img src="/images/equations/RiemannZetaFunction/Inline163.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="72" height="21" alt="zeta(2n+1)" /> is irrational, and subsequently that at least one of <img src="/images/equations/RiemannZetaFunction/Inline164.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(5)" />, <img src="/images/equations/RiemannZetaFunction/Inline165.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(7)" />, ..., <img src="/images/equations/RiemannZetaFunction/Inline166.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="40" height="21" alt="zeta(21)" /> is <a href="/IrrationalNumber.html">irrational</a> (Rivoal 2001). This result was subsequently tightened by Zudilin (2001), who showed that at least one of <img src="/images/equations/RiemannZetaFunction/Inline167.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(5)" />, <img src="/images/equations/RiemannZetaFunction/Inline168.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(7)" />, <img src="/images/equations/RiemannZetaFunction/Inline169.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(9)" />, or <img src="/images/equations/RiemannZetaFunction/Inline170.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="40" height="21" alt="zeta(11)" /> is <a href="/IrrationalNumber.html">irrational</a>. </p> <p> A number of interesting sums for <img src="/images/equations/RiemannZetaFunction/Inline171.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(n)" />, with <img src="/images/equations/RiemannZetaFunction/Inline172.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> a <a href="/PositiveInteger.html">positive integer</a>, can be written in terms of binomial coefficients as the <a href="/BinomialSums.html">binomial sums</a> </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline173.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(2)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline174.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline175.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="98" height="67" alt="3sum_(k=1)^(infty)1/(k^2(2k; k))" /></td><td align="right" width="10"> <div id="eqn48" class="eqnum"> (48) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline176.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(3)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline177.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline178.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="104" height="68" alt="5/2sum_(k=1)^(infty)((-1)^(k-1))/(k^3(2k; k))" /></td><td align="right" width="10"> <div id="eqn49" class="eqnum"> (49) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline179.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(4)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline180.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline181.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="112" height="67" alt="(36)/(17)sum_(k=1)^(infty)1/(k^4(2k; k))" /></td><td align="right" width="10"> <div id="eqn50" class="eqnum"> (50) </div> </td></tr> </table> </div> <p> (Guy 1994, p. 257; Bailey <i>et al. </i>2007, p. 70). Apéry arrived at his result with the aid of the <img src="/images/equations/RiemannZetaFunction/Inline182.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="24" height="21" alt="k^(-3)" /> sum formula above. A relation <a href="/OftheForm.html">of the form</a> </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation27.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="156" height="68" alt=" zeta(5)=Z_5sum_(k=1)^infty((-1)^(k-1))/(k^5(2k; k)) " /></td><td align="right" width="3"> <div id="eqn51" class="eqnum"> (51) </div> </td></tr> </table> </div> <p> has been searched for with <img src="/images/equations/RiemannZetaFunction/Inline183.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="17" height="22" alt="Z_5" /> a <a href="/RationalNumber.html">rational</a> or <a href="/AlgebraicNumber.html">algebraic number</a>, but if <img src="/images/equations/RiemannZetaFunction/Inline184.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="17" height="22" alt="Z_5" /> is a <a href="/Root.html">root</a> of a <a href="/Polynomial.html">polynomial</a> of degree 25 or less, then the Euclidean norm of the coefficients must be larger than <img src="/images/equations/RiemannZetaFunction/Inline185.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="87" height="21" alt="1.24×10^(383)" />, and if <img src="/images/equations/RiemannZetaFunction/Inline186.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(5)" /> if algebraic of degree 25 or less, then the norm of coefficients must exceed <img src="/images/equations/RiemannZetaFunction/Inline187.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="87" height="21" alt="1.98×10^(380)" /> (Bailey <i>et al. </i>2007, pp. 70-71, updating Bailey and Plouffe). Therefore, no such sums for <img src="/images/equations/RiemannZetaFunction/Inline188.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(n)" /> are known for <img src="/images/equations/RiemannZetaFunction/Inline189.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="n>=5" />. </p> <p> The identity </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline190.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="76" height="50" alt="sum_(k=1)^(infty)1/(k^2-x^2)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline191.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline192.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="117" height="49" alt="sum_(n=0)^(infty)zeta(2n+2)x^(2n)" /></td><td align="right" width="10"> <div id="eqn52" class="eqnum"> (52) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline193.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline194.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline195.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="106" height="42" alt="(1-pixcot(pix))/(2x^2)" /></td><td align="right" width="10"> <div id="eqn53" class="eqnum"> (53) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline196.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline197.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline198.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="241" height="78" alt="3sum_(k=1)^(infty)1/(k^2(2k; k)(1-(x^2)/(k^2)))product_(m=1)^(k-1)(1-(4x^2)/(m^2))/(1-(x^2)/(m^2))" /></td><td align="right" width="10"> <div id="eqn54" class="eqnum"> (54) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline199.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline200.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline201.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="320" height="55" alt="(3_4F_3(1,2,1-2x,1+2x;3/2,2-x,2+x;1/4))/(2(1-x^2))" /></td><td align="right" width="10"> <div id="eqn55" class="eqnum"> (55) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/RiemannZetaFunction/Inline202.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="x" /> is complex number not equal to a nonzero integer gives an Apéry-like formula for even positive <img src="/images/equations/RiemannZetaFunction/Inline203.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> (Bailey <i>et al. </i>2006, pp. 72-77). </p> <p> The Riemann zeta function <img src="/images/equations/RiemannZetaFunction/Inline204.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="44" height="21" alt="zeta(2n)" /> may be computed analytically for <a href="/EvenNumber.html">even</a> <img src="/images/equations/RiemannZetaFunction/Inline205.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> using either <a href="/ContourIntegral.html">contour integration</a> or <a href="/ParsevalsTheorem.html">Parseval's theorem</a> with the appropriate <a href="/FourierSeries.html">Fourier series</a>. An unexpected and important formula involving a product over the <a href="/PrimeNumber.html">primes</a> was first discovered by Euler in 1737, </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline206.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="88" height="20" alt="zeta(s)(1-2^(-s))" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline207.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline208.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="188" height="40" alt="(1+1/(2^s)+1/(3^s)+...)(1-1/(2^s))" /></td><td align="right" width="10"> <div id="eqn56" class="eqnum"> (56) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline209.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline210.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline211.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="282" height="40" alt="(1+1/(2^s)+1/(3^s)+...)-(1/(2^s)+1/(4^s)+1/(6^s)+...)" /></td><td align="right" width="10"> <div id="eqn57" class="eqnum"> (57) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline212.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="148" height="20" alt="zeta(s)(1-2^(-s))(1-3^(-s))" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline213.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline214.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="327" height="40" alt="(1+1/(3^s)+1/(5^s)+1/(7^s)+...)-(1/(3^s)+1/(9^s)+1/(15^s)+...)" /></td><td align="right" width="10"> <div id="eqn58" class="eqnum"> (58) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline215.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="247" height="22" alt="zeta(s)(1-2^(-s))(1-3^(-s))...(1-p_n^(-s))..." /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline216.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline217.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="113" height="49" alt="zeta(s)product_(n=1)^(infty)(1-p_n^(-s))" /></td><td align="right" width="10"> <div id="eqn59" class="eqnum"> (59) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline218.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline219.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline220.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="13" height="20" alt="1." /></td><td align="right" width="10"> <div id="eqn60" class="eqnum"> (60) </div> </td></tr> </table> </div> <p> Here, each subsequent multiplication by the <img src="/images/equations/RiemannZetaFunction/Inline221.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />th <a href="/PrimeNumber.html">prime</a> <img src="/images/equations/RiemannZetaFunction/Inline222.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="18" height="22" alt="p_n" /> leaves only terms that are <a href="/Power.html">powers</a> of <img src="/images/equations/RiemannZetaFunction/Inline223.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="25" height="22" alt="p^(-s)" />. Therefore, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation28.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="161" height="53" alt=" zeta(s)=[product_(n=1)^infty(1-p_n^(-s))]^(-1), " /></td><td align="right" width="3"> <div id="eqn61" class="eqnum"> (61) </div> </td></tr> </table> </div> <p> which is known as the <a href="/EulerProduct.html">Euler product</a> formula (Hardy 1999, p. 18; Krantz 1999, p. 159), and called "the golden key" by Derbyshire (2004, pp. 104-106). The formula can also be written </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation29.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="362" height="61" alt=" zeta(s)=(1-2^(-s))^(-1)product_(q=1; (mod 4))(1-q^(-s))^(-1)product_(r=3; (mod 4))(1-r^(-s))^(-1), " /></td><td align="right" width="3"> <div id="eqn62" class="eqnum"> (62) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline224.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="q" /> and <img src="/images/equations/RiemannZetaFunction/Inline225.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="7" height="21" alt="r" /> are the primes congruent to 1 and 3 modulo 4, respectively. </p> <p> For <a href="/EvenNumber.html">even</a> <img src="/images/equations/RiemannZetaFunction/Inline226.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="n>=2" />, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation30.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="137" height="42" alt=" zeta(n)=(2^(n-1)|B_n|pi^n)/(n!), " /></td><td align="right" width="3"> <div id="eqn63" class="eqnum"> (63) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline227.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="18" height="21" alt="B_n" /> is a <a href="/BernoulliNumber.html">Bernoulli number</a> (Mathews and Walker 1970, pp. 50-53; Havil 2003, p. 194). Another intimate connection with the <a href="/BernoulliNumber.html">Bernoulli numbers</a> is provided by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation31.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="159" height="20" alt=" B_n=(-1)^(n+1)nzeta(1-n) " /></td><td align="right" width="3"> <div id="eqn64" class="eqnum"> (64) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/RiemannZetaFunction/Inline228.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="n>=1" />, which can be written </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation32.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="117" height="20" alt=" B_n=-nzeta(1-n) " /></td><td align="right" width="3"> <div id="eqn65" class="eqnum"> (65) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/RiemannZetaFunction/Inline229.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="n>=2" />. (In both cases, only the even cases are of interest since <img src="/images/equations/RiemannZetaFunction/Inline230.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="48" height="21" alt="B_n=0" /> trivially for odd <img src="/images/equations/RiemannZetaFunction/Inline231.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />.) Rewriting (<a href="#eqn65">65</a>), </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation33.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="113" height="39" alt=" zeta(-n)=-(B_(n+1))/(n+1) " /></td><td align="right" width="3"> <div id="eqn66" class="eqnum"> (66) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/RiemannZetaFunction/Inline232.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="n=1" />, 3, ... (Havil 2003, p. 194), where <img src="/images/equations/RiemannZetaFunction/Inline233.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="18" height="21" alt="B_n" /> is a <a href="/BernoulliNumber.html">Bernoulli number</a>, the first few values of which are <img src="/images/equations/RiemannZetaFunction/Inline234.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="50" height="21" alt="-1/12" />, 1/120, <img src="/images/equations/RiemannZetaFunction/Inline235.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="59" height="21" alt="-1/252" />, 1/240, ... (OEIS <a href="http://oeis.org/A001067">A001067</a> and <a href="http://oeis.org/A006953">A006953</a>). </p> <p> Although no analytic form for <img src="/images/equations/RiemannZetaFunction/Inline236.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(n)" /> is known for <a href="/OddNumber.html">odd</a> <img src="/images/equations/RiemannZetaFunction/Inline237.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation34.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="122" height="50" alt=" zeta(3)=1/2sum_(k=1)^infty(H_k)/(k^2), " /></td><td align="right" width="3"> <div id="eqn67" class="eqnum"> (67) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline238.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="H_k" /> is a <a href="/HarmonicNumber.html">harmonic number</a> (Stark 1974). In addition, <img src="/images/equations/RiemannZetaFunction/Inline239.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(n)" /> can be expressed as the sum limit </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation35.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="279" height="51" alt=" zeta(n)=lim_(x->infty)1/((2x+1)^n)sum_(k=1)^x[cot(k/(2x+1))]^n " /></td><td align="right" width="3"> <div id="eqn68" class="eqnum"> (68) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/RiemannZetaFunction/Inline240.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="n=3" />, 5, ... (Apostol 1973, given incorrectly in Stark 1974). </p> <p> For <img src="/images/equations/RiemannZetaFunction/Inline241.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="33" height="21" alt="mu(n)" /> the <a href="/MoebiusFunction.html">Möbius function</a>, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation36.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="116" height="49" alt=" 1/(zeta(s))=sum_(n=1)^infty(mu(n))/(n^s) " /></td><td align="right" width="3"> <div id="eqn69" class="eqnum"> (69) </div> </td></tr> </table> </div> <p> (Havil 2003, p. 209). </p> <p> The values of <img src="/images/equations/RiemannZetaFunction/Inline242.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(n)" /> for small positive integer values of <img src="/images/equations/RiemannZetaFunction/Inline243.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> are </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline244.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(1)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline245.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline246.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="14" height="20" alt="infty" /></td><td align="right" width="10"> <div id="eqn70" class="eqnum"> (70) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline247.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(2)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline248.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline249.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="21" height="42" alt="(pi^2)/6" /></td><td align="right" width="10"> <div id="eqn71" class="eqnum"> (71) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline250.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(3)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline251.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline252.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="117" height="20" alt="1.2020569032..." /></td><td align="right" width="10"> <div id="eqn72" class="eqnum"> (72) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline253.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(4)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline254.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline255.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="23" height="42" alt="(pi^4)/(90)" /></td><td align="right" width="10"> <div id="eqn73" class="eqnum"> (73) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline256.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(5)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline257.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline258.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="117" height="20" alt="1.0369277551..." /></td><td align="right" width="10"> <div id="eqn74" class="eqnum"> (74) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline259.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(6)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline260.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline261.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="32" height="42" alt="(pi^6)/(945)" /></td><td align="right" width="10"> <div id="eqn75" class="eqnum"> (75) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline262.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(7)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline263.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline264.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="117" height="20" alt="1.0083492774..." /></td><td align="right" width="10"> <div id="eqn76" class="eqnum"> (76) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline265.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(8)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline266.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline267.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="40" height="42" alt="(pi^8)/(9450)" /></td><td align="right" width="10"> <div id="eqn77" class="eqnum"> (77) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline268.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(9)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline269.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline270.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="117" height="20" alt="1.0020083928..." /></td><td align="right" width="10"> <div id="eqn78" class="eqnum"> (78) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline271.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="38" height="20" alt="zeta(10)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline272.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline273.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="55" height="42" alt="(pi^(10))/(93555)." /></td><td align="right" width="10"> <div id="eqn79" class="eqnum"> (79) </div> </td></tr> </table> </div> <p> Euler gave <img src="/images/equations/RiemannZetaFunction/Inline274.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(2)" /> to <img src="/images/equations/RiemannZetaFunction/Inline275.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="40" height="21" alt="zeta(26)" /> for <a href="/EvenNumber.html">even</a> <img src="/images/equations/RiemannZetaFunction/Inline276.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> (Wells 1986, p. 54), and Stieltjes (1993) determined the values of <img src="/images/equations/RiemannZetaFunction/Inline277.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(2)" />, ..., <img src="/images/equations/RiemannZetaFunction/Inline278.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="40" height="21" alt="zeta(70)" /> to 30 digits of accuracy in 1887. The denominators of <img src="/images/equations/RiemannZetaFunction/Inline279.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="44" height="21" alt="zeta(2n)" /> for <img src="/images/equations/RiemannZetaFunction/Inline280.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="n=1" />, 2, ... are 6, 90, 945, 9450, 93555, 638512875, ... (OEIS <a href="http://oeis.org/A002432">A002432</a>). The numbers of decimal digits in the denominators of <img src="/images/equations/RiemannZetaFunction/Inline281.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="47" height="21" alt="zeta(10^n)" /> for <img src="/images/equations/RiemannZetaFunction/Inline282.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="n=0" />, 1, ... are 1, 5, 133, 2277, 32660, 426486, 5264705, ... (OEIS <a href="http://oeis.org/A114474">A114474</a>). </p> <p> An integral for positive even integers is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation37.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="315" height="50" alt=" zeta(2n)=((-1)^(n+1)2^(2n-3)pi^(2n))/((2^(2n)-1)(2n-2)!)int_0^1E_(2(n-1))(x)dx, " /></td><td align="right" width="3"> <div id="eqn80" class="eqnum"> (80) </div> </td></tr> </table> </div> <p> and integrals for positive odd integers are given by </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline283.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="67" height="20" alt="zeta(2n+1)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline284.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline285.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="289" height="50" alt="((-1)^n2^(2n-1)pi^(2n+1))/((2^(2n+1)-1)(2n)!)int_0^1E_(2n)(x)tan(1/2pix)dx" /></td><td align="right" width="10"> <div id="eqn81" class="eqnum"> (81) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline286.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline287.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline288.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="289" height="50" alt="((-1)^n2^(2n-1)pi^(2n+1))/((2^(2n+1)-1)(2n)!)int_0^1E_(2n)(x)cot(1/2pix)dx" /></td><td align="right" width="10"> <div id="eqn82" class="eqnum"> (82) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline289.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline290.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline291.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="289" height="45" alt="((-1)^n2^(2n)pi^(2n+1))/((2n+1)!)int_0^1B_(2n+1)(x)tan(1/2pix)dx" /></td><td align="right" width="10"> <div id="eqn83" class="eqnum"> (83) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline292.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline293.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline294.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="306" height="45" alt="((-1)^(n+1)2^(2n)pi^(2n+1))/((2n+1)!)int_0^1B_(2n+1)(x)cot(1/2pix)dx," /></td><td align="right" width="10"> <div id="eqn84" class="eqnum"> (84) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline295.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="42" height="21" alt="E_n(x)" /> is an <a href="/EulerPolynomial.html">Euler polynomial</a> and <img src="/images/equations/RiemannZetaFunction/Inline296.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="41" height="21" alt="B_n(x)" /> is a <a href="/BernoulliPolynomial.html">Bernoulli polynomial</a> (Cvijović and Klinowski 2002; J. Crepps, pers. comm., Apr. 2002). </p> <p> The value of <img src="/images/equations/RiemannZetaFunction/Inline297.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(0)" /> can be computed by performing the inner sum in equation (◇) with <img src="/images/equations/RiemannZetaFunction/Inline298.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="s=0" />, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation38.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="219" height="51" alt=" zeta(0)=-sum_(n=0)^infty1/(2^(n+1))sum_(k=0)^n(-1)^k(n; k), " /></td><td align="right" width="3"> <div id="eqn85" class="eqnum"> (85) </div> </td></tr> </table> </div> <p> to obtain </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation39.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="239" height="49" alt=" zeta(0)=-sum_(n=0)^infty(delta_(0,n))/(2^(n+1))=-1/(2^(0+1))=-1/2, " /></td><td align="right" width="3"> <div id="eqn86" class="eqnum"> (86) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline299.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="24" height="23" alt="delta_(0,n)" /> is the <a href="/KroneckerDelta.html">Kronecker delta</a>. </p> <p> Similarly, the value of <img src="/images/equations/RiemannZetaFunction/Inline300.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="43" height="21" alt="zeta(-1)" /> can be computed by performing the inner sum in equation (◇) with <img src="/images/equations/RiemannZetaFunction/Inline301.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="49" height="21" alt="s=-1" />, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation40.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="294" height="51" alt=" zeta(-1)=-1/3sum_(n=0)^infty1/(2^(n+1))sum_(k=0)^n(-1)^k(n; k)(k+1), " /></td><td align="right" width="3"> <div id="eqn87" class="eqnum"> (87) </div> </td></tr> </table> </div> <p> which gives </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline302.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="41" height="20" alt="zeta(-1)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline303.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline304.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="133" height="49" alt="-1/3sum_(n=0)^(infty)(delta_(0,n)-ndelta_(1,n))/(2^(n+1))" /></td><td align="right" width="10"> <div id="eqn88" class="eqnum"> (88) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline305.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline306.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline307.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="127" height="41" alt="-1/3(1/(2^(0+1))-1/(2^(1+1)))" /></td><td align="right" width="10"> <div id="eqn89" class="eqnum"> (89) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline308.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline309.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline310.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="38" height="39" alt="-1/(12)." /></td><td align="right" width="10"> <div id="eqn90" class="eqnum"> (90) </div> </td></tr> </table> </div> <p> This value is related to a deep result in renormalization theory (Elizalde <i>et al. </i>1994, 1995, Bloch 1996, Lepowski 1999). </p> <p> It is apparently not known if the value </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation41.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="190" height="26" alt=" zeta(1/2)=-1.46035450880... " /></td><td align="right" width="3"> <div id="eqn91" class="eqnum"> (91) </div> </td></tr> </table> </div> <p> (OEIS <a href="http://oeis.org/A059750">A059750</a>) can be expressed in terms of known mathematical constants. This constant appears, for example, in <a href="/KnuthsSeries.html">Knuth's series</a>. </p> <p> Rapidly converging series for <img src="/images/equations/RiemannZetaFunction/Inline311.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(n)" /> for <img src="/images/equations/RiemannZetaFunction/Inline312.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> odd were first discovered by Ramanujan (Zucker 1979, 1984, Berndt 1988, Bailey <i>et al. </i>1997, Cohen 2000). For <img src="/images/equations/RiemannZetaFunction/Inline313.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="n>1" /> and <img src="/images/equations/RiemannZetaFunction/Inline314.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="96" height="21" alt="n=3 (mod 4)" />, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation42.svg" data-src-small="/images/equations/RiemannZetaFunction/NumberedEquation42_400.svg" data-src-default="/images/equations/RiemannZetaFunction/NumberedEquation42.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="485" height="54" data-big="485 54" data-small="296 115" border="0" alt=" zeta(n)=(2^(n-1)pi^n)/((n+1)!)sum_(k=0)^((n+1)/2)(-1)^(k-1)(n+1; 2k)B_(n+1-2k)B_(2k)-2sum_(k=1)^infty1/(k^n(e^(2pik)-1)), " /></td><td align="right" width="3"> <div id="eqn92" class="eqnum"> (92) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline315.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="18" height="21" alt="B_k" /> is again a <a href="/BernoulliNumber.html">Bernoulli number</a> and <img src="/images/equations/RiemannZetaFunction/Inline316.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="43" alt="(n; k)" /> is a <a href="/BinomialCoefficient.html">binomial coefficient</a>. The values of the left-hand sums (divided by <img src="/images/equations/RiemannZetaFunction/Inline317.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="pi^n" />) in (<a href="#eqn92">92</a>) for <img src="/images/equations/RiemannZetaFunction/Inline318.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="n=3" />, 7, 11, ... are 7/180, 19/56700, 1453/425675250, 13687/390769879500, 7708537/21438612514068750, ... (OEIS <a href="http://oeis.org/A057866">A057866</a> and <a href="http://oeis.org/A057867">A057867</a>). For <img src="/images/equations/RiemannZetaFunction/Inline319.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="n>=5" /> and <img src="/images/equations/RiemannZetaFunction/Inline320.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="96" height="21" alt="n=1 (mod 4)" />, the corresponding formula is slightly messier, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation43.svg" data-src-small="/images/equations/RiemannZetaFunction/NumberedEquation43_400.svg" data-src-default="/images/equations/RiemannZetaFunction/NumberedEquation43.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="636" height="59" data-big="636 59" data-small="330 125" border="0" alt=" zeta(n)=((2pi)^n)/((n+1)!(n-1))sum_(k=0)^((n+1)/4)(-1)^k(n+1-4k)(n+1; 2k)B_(n+1-2k)B_(2k)-2sum_(k=1)^infty(e^(2pik)(1+(4pik)/(n-1))-1)/(k^n(e^(2pik)-1)^2) " /></td><td align="right" width="3"> <div id="eqn93" class="eqnum"> (93) </div> </td></tr> </table> </div> <p> (Cohen 2000). </p> <p> Defining </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation44.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="174" height="50" alt=" S_+/-(n)=sum_(k=1)^infty1/(k^n(e^(2pik)+/-1)), " /></td><td align="right" width="3"> <div id="eqn94" class="eqnum"> (94) </div> </td></tr> </table> </div> <p> the first few values can then be written </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline321.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(3)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline322.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline323.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="108" height="26" alt="7/(180)pi^3-2S_-(3)" /></td><td align="right" width="10"> <div id="eqn95" class="eqnum"> (95) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline324.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(5)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline325.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline326.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="192" height="26" alt="1/(294)pi^5-(72)/(35)S_-(5)-2/(35)S_+(5)" /></td><td align="right" width="10"> <div id="eqn96" class="eqnum"> (96) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline327.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(7)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline328.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline329.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="122" height="26" alt="(19)/(56700)pi^7-2S_-(7)" /></td><td align="right" width="10"> <div id="eqn97" class="eqnum"> (97) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline330.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(9)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline331.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline332.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="232" height="26" alt="(125)/(3704778)pi^9-(992)/(495)S_-(9)-2/(495)S_+(9)" /></td><td align="right" width="10"> <div id="eqn98" class="eqnum"> (98) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline333.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="38" height="20" alt="zeta(11)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline334.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline335.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="162" height="26" alt="(1453)/(425675250)pi^(11)-2S_-(11)" /></td><td align="right" width="10"> <div id="eqn99" class="eqnum"> (99) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline336.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="38" height="20" alt="zeta(13)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline337.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline338.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="286" height="26" alt="(89)/(257432175)pi^(13)-(16512)/(8255)S_-(13)-2/(8255)S_+(13)" /></td><td align="right" width="10"> <div id="eqn100" class="eqnum"> (100) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline339.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="38" height="20" alt="zeta(15)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline340.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline341.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="182" height="26" alt="(13687)/(390769879500)pi^(15)-2S_-(15)" /></td><td align="right" width="10"> <div id="eqn101" class="eqnum"> (101) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline342.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="38" height="20" alt="zeta(17)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline343.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline344.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="346" height="26" alt="(397549)/(112024529867250)pi^(17)-(261632)/(130815)S_-(17)-2/(130815)S_+(17)" /></td><td align="right" width="10"> <div id="eqn102" class="eqnum"> (102) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline345.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="38" height="20" alt="zeta(19)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline346.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline347.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="216" height="26" alt="(7708537)/(21438612514068750)pi^(19)-2S_-(19)" /></td><td align="right" width="10"> <div id="eqn103" class="eqnum"> (103) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline348.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="38" height="20" alt="zeta(21)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline349.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline350.svg" data-src-small="/images/equations/RiemannZetaFunction/Inline350_400.svg" data-src-default="/images/equations/RiemannZetaFunction/Inline350.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="409" height="26" data-big="409 26" data-small="232 56" border="0" alt="(68529640373)/(1881063815762259253125)pi^(21)-(4196352)/(2098175)S_-(21)-2/(2098175)S_+(21)" /></td><td align="right" width="10"> <div id="eqn104" class="eqnum"> (104) </div> </td></tr> </table> </div> <p> (Plouffe 1998). </p> <p> Another set of related formulas are </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline351.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(3)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline352.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline353.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="321" height="50" alt="(pi^3)/(28)+(16)/7sum_(n=1)^(infty)1/(n^3(e^(npi)+1))-2/7sum_(n=1)^(infty)1/(n^3(e^(2pin)+1))" /></td><td align="right" width="10"> <div id="eqn105" class="eqnum"> (105) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline354.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(5)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline355.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline356.svg" data-src-small="/images/equations/RiemannZetaFunction/Inline356_400.svg" data-src-default="/images/equations/RiemannZetaFunction/Inline356.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="450" height="49" data-big="450 49" data-small="286 108" border="0" alt="24sum_(n=1)^(infty)1/(n^5(e^(npi)-1))-(259)/(10)sum_(n=1)^(infty)1/(n^5(e^(2pin)-1))-1/(10)sum_(n=1)^(infty)1/(n^5(e^(4pin)-1))" /></td><td align="right" width="10"> <div id="eqn106" class="eqnum"> (106) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline357.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(5)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline358.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline359.svg" data-src-small="/images/equations/RiemannZetaFunction/Inline359_400.svg" data-src-default="/images/equations/RiemannZetaFunction/Inline359.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="895" height="50" data-big="895 50" data-small="352 168" border="0" alt="-(7pi^5)/(1840)+(328)/(115)sum_(n=1)^(infty)1/(n^5(e^(pin)-1))-(419)/(460)sum_(n=1)^(infty)1/(n^5(e^(2pin)-1))-9/(115)sum_(n=1)^(infty)1/(n^5(e^(3pin)-1))+(261)/(1840)sum_(n=1)^(infty)1/(n^5(e^(6pin)-1))-9/(1840)sum_(n=1)^(infty)1/(n^5(e^(12pin)-1))" /></td><td align="right" width="10"> <div id="eqn107" class="eqnum"> (107) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline360.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(7)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline361.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline362.svg" data-src-small="/images/equations/RiemannZetaFunction/Inline362_400.svg" data-src-default="/images/equations/RiemannZetaFunction/Inline362.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="465" height="49" data-big="465 49" data-small="286 108" border="0" alt="(304)/(13)sum_(n=1)^(infty)1/(n^7(e^(pin)-1))-(103)/4sum_(n=1)^(infty)1/(n^7(e^(2pin)-1))-(19)/(52)sum_(n=1)^(infty)1/(n^7(e^(4pin)-1))" /></td><td align="right" width="10"> <div id="eqn108" class="eqnum"> (108) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline363.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(9)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline364.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline365.svg" data-src-small="/images/equations/RiemannZetaFunction/Inline365_400.svg" data-src-default="/images/equations/RiemannZetaFunction/Inline365.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="945" height="49" data-big="945 49" data-small="336 167" border="0" alt="(64)/3sum_(n=1)^(infty)1/(n^9(e^(pin)-1))+(441)/(20)sum_(n=1)^(infty)1/(n^9(e^(2pin)-1))-32sum_(n=1)^(infty)1/(n^9(e^(3pin)-1))-(4763)/(60)sum_(n=1)^(infty)1/(n^9(e^(4pin)-1))+(529)/8sum_(n=1)^(infty)1/(n^9(e^(6pin)-1))-1/8sum_(n=1)^(infty)1/(n^9(e^(12pin)-1))" /></td><td align="right" width="10"> <div id="eqn109" class="eqnum"> (109) </div> </td></tr> </table> </div> <p> (Plouffe 2006). </p> <p> Multiterm sums for odd <img src="/images/equations/RiemannZetaFunction/Inline366.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(n)" /> include </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline367.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(5)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline368.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline369.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="247" height="71" alt="2sum_(k=1)^(infty)((-1)^(k+1))/(k^5(2k; k))-5/2sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((2)))/(k^3(2k; k))" /></td><td align="right" width="10"> <div id="eqn110" class="eqnum"> (110) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline370.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(7)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline371.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline372.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="261" height="71" alt="5/2sum_(k=1)^(infty)((-1)^(k+1))/(k^7(2k; k))+(25)/2sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((4)))/(k^3(2k; k))" /></td><td align="right" width="10"> <div id="eqn111" class="eqnum"> (111) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline373.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="zeta(9)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline374.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline375.svg" data-src-small="/images/equations/RiemannZetaFunction/Inline375_400.svg" data-src-default="/images/equations/RiemannZetaFunction/Inline375.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="745" height="71" data-big="745 71" data-small="321 241" border="0" alt="9/4sum_(k=1)^(infty)((-1)^(k+1))/(k^9(2k; k))-5/4sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((2)))/(k^7(2k; k))+5sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((4)))/(k^5(2k; k))+(45)/4sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((6)))/(k^3(2k; k))-(25)/4sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((2))H_(k-1)^((4)))/(k^3(2k; k))" /></td><td align="right" width="10"> <div id="eqn112" class="eqnum"> (112) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline376.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="38" height="20" alt="zeta(11)" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline377.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline378.svg" data-src-small="/images/equations/RiemannZetaFunction/Inline378_400.svg" data-src-default="/images/equations/RiemannZetaFunction/Inline378.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="607" height="74" data-big="607 74" data-small="312 161" border="0" alt="5/2sum_(k=1)^(infty)((-1)^(k+1))/(k^(11)(2k; k))+(25)/2sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((4)))/(k^7(2k; k))-(75)/4sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((8)))/(k^3(2k; k))+(125)/4sum_(k=1)^(infty)((-1)^(k+1)[H_(k-1)^((4))]^2)/(k^3(2k; k))" /></td><td align="right" width="10"> <div id="eqn113" class="eqnum"> (113) </div> </td></tr> </table> </div> <p> (Borwein and Bradley 1996, 1997; Bailey <i>et al. </i>2007, p. 71), where <img src="/images/equations/RiemannZetaFunction/Inline379.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="H_n^((r))" /> is a generalized <a href="/HarmonicNumber.html">harmonic number</a>. </p> <p> G. Huvent (2002) found the beautiful formula </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation45.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="228" height="49" alt=" zeta(5)=-(16)/(11)sum_(n=1)^infty([2(-1)^n+1]H_n)/(n^4). " /></td><td align="right" width="3"> <div id="eqn114" class="eqnum"> (114) </div> </td></tr> </table> </div> <p> A number of sum identities involving <img src="/images/equations/RiemannZetaFunction/Inline380.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(n)" /> include </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline381.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="89" height="49" alt="sum_(n=2)^(infty)[zeta(n)-1]" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline382.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline383.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="9" height="20" alt="1" /></td><td align="right" width="10"> <div id="eqn115" class="eqnum"> (115) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline384.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="112" height="51" alt="sum_(n=2,4,...)^(infty)[zeta(n)-1]" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline385.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline386.svg" class="displayformula" style="max-width:100%;max-height:100%;" width="12" height="26" border="0" alt="3/4" /></td><td align="right" width="10"> <div id="eqn116" class="eqnum"> (116) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline387.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="112" height="51" alt="sum_(n=3,5,...)^(infty)[zeta(n)-1]" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline388.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline389.svg" class="displayformula" style="max-width:100%;max-height:100%;" width="12" height="26" border="0" alt="1/4" /></td><td align="right" width="10"> <div id="eqn117" class="eqnum"> (117) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline390.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="129" height="49" alt="sum_(n=2)^(infty)(-1)^n[zeta(n)-1]" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline391.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline392.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="16" height="26" alt="1/2." /></td><td align="right" width="10"> <div id="eqn118" class="eqnum"> (118) </div> </td></tr> </table> </div> <p> Sums involving integers multiples of the argument include </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline393.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="101" height="49" alt="sum_(n=1)^(infty)[zeta(2n)-1]" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline394.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline395.svg" class="displayformula" style="max-width:100%;max-height:100%;" width="12" height="26" border="0" alt="3/4" /></td><td align="right" width="10"> <div id="eqn119" class="eqnum"> (119) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline396.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="101" height="49" alt="sum_(n=1)^(infty)[zeta(3n)-1]" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline397.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline398.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="288" height="31" alt="1/3[-(-1)^(2/3)H_((3-isqrt(3))/2)+(-1)^(1/3)H_((3+isqrt(3))/2)]" /></td><td align="right" width="10"> <div id="eqn120" class="eqnum"> (120) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline399.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="101" height="49" alt="sum_(n=1)^(infty)[zeta(4n)-1]" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline400.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline401.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="106" height="26" alt="1/8(7-2cothpi)," /></td><td align="right" width="10"> <div id="eqn121" class="eqnum"> (121) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline402.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="H_n" /> is a <a href="/HarmonicNumber.html">harmonic number</a>. </p> <p> Two surprising sums involving <img src="/images/equations/RiemannZetaFunction/Inline403.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(x)" /> are given by </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline404.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="99" height="51" alt="sum_(k=2)^(infty)((-1)^kzeta(k))/k" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline405.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline406.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="8" height="20" alt="gamma" /></td><td align="right" width="10"> <div id="eqn122" class="eqnum"> (122) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline407.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="84" height="50" alt="sum_(k=2)^(infty)(zeta(k)-1)/k" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline408.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline409.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="37" height="20" alt="1-gamma," /></td><td align="right" width="10"> <div id="eqn123" class="eqnum"> (123) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline410.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="gamma" /> is the <a href="/Euler-MascheroniConstant.html">Euler-Mascheroni constant</a> (Havil 2003, pp. 109 and 111-112). Equation (<a href="#eqn122">122</a>) can be generalized to </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation46.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="195" height="51" alt=" sum_(k=2)^infty((-x)^kzeta(k))/k=xgamma+ln(x!) " /></td><td align="right" width="3"> <div id="eqn124" class="eqnum"> (124) </div> </td></tr> </table> </div> <p> (T. Drane, pers. comm., Jul. 7, 2006) for <img src="/images/equations/RiemannZetaFunction/Inline411.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="77" height="21" alt="-1<x<=1" />. </p> <p> Other unexpected sums are </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation47.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="195" height="49" alt=" sum_(n=1)^infty(zeta(2n))/(n(2n+1)2^(2n))=lnpi-1 " /></td><td align="right" width="3"> <div id="eqn125" class="eqnum"> (125) </div> </td></tr> </table> </div> <p> (Tyler and Chernhoff 1985; Boros and Moll 2004, p. 248) and </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation48.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="194" height="49" alt=" sum_(n=1)^infty(zeta(2n))/(n(2n+1))=ln(2pi)-1. " /></td><td align="right" width="3"> <div id="eqn126" class="eqnum"> (126) </div> </td></tr> </table> </div> <p> (<a href="#eqn125">125</a>) is a special case of </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation49.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="353" height="83" alt=" sum_(k=1)^infty(zeta(2k,z))/(k(2k+1)2^(2k)) =(2z-1)ln(z-1/2)-2z+1+ln(2pi)-2lnGamma(z), " /></td><td align="right" width="3"> <div id="eqn127" class="eqnum"> (127) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline412.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="48" height="21" alt="zeta(s,a)" /> is a <a href="/HurwitzZetaFunction.html">Hurwitz zeta function</a> (Danese 1967; Boros and Moll 2004, p. 248). </p> <p> Considering the sum </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation50.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="156" height="54" alt=" S_n=sum_(k=2)^(n-2)(zeta(k)zeta(n-k))/(2^k), " /></td><td align="right" width="3"> <div id="eqn128" class="eqnum"> (128) </div> </td></tr> </table> </div> <p> then </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation51.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="92" height="30" alt=" lim_(n->infty)S_n=ln2, " /></td><td align="right" width="3"> <div id="eqn129" class="eqnum"> (129) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline413.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="27" height="21" alt="ln2" /> is the <a href="/NaturalLogarithmof2.html">natural logarithm of 2</a>, which is a particular case of </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation52.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="304" height="54" alt=" lim_(n->infty)sum_(k=2)^(n-2)zeta(k)zeta(n-k)x^(k-1)=x^(-1)-psi_0(-x)-gamma, " /></td><td align="right" width="3"> <div id="eqn130" class="eqnum"> (130) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline414.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="22" alt="psi_0(z)" /> is the <a href="/DigammaFunction.html">digamma function</a> and <img src="/images/equations/RiemannZetaFunction/Inline415.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="gamma" /> is the <a href="/Euler-MascheroniConstant.html">Euler-Mascheroni constant</a>, which can be derived from </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation53.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="203" height="50" alt=" sum_(k=2)^inftyzeta(k)x^(k-1)=-psi_0(1-x)-gamma " /></td><td align="right" width="3"> <div id="eqn131" class="eqnum"> (131) </div> </td></tr> </table> </div> <p> (B. Cloitre, pers. comm., Dec. 11, 2005; cf. Borwein <i>et al. </i>2000, eqn. 27). </p> <p> A generalization of a result of Ramanujan (who gave the <img src="/images/equations/RiemannZetaFunction/Inline416.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="43" height="21" alt="m=1" /> case) is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/RiemannZetaFunction/NumberedEquation54.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="344" height="51" alt=" sum_(k=1)^infty1/([k(k+1)]^(2m+1))=-2sum_(k=0)^mzeta(2k)(4m-2k+1; 2m), " /></td><td align="right" width="3"> <div id="eqn132" class="eqnum"> (132) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/RiemannZetaFunction/Inline417.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="43" alt="(n; k)" /> is a <a href="/BinomialCoefficient.html">binomial coefficient</a> (B. Cloitre, pers. comm., Sep. 20, 2005). </p> <p> An additional set of sums over <img src="/images/equations/RiemannZetaFunction/Inline418.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(n)" /> is given by </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline419.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="18" height="20" alt="C_1" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline420.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline421.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="60" height="49" alt="sum_(n=2)^(infty)(zeta(n))/(n!)" /></td><td align="right" width="10"> <div id="eqn133" class="eqnum"> (133) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline422.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline423.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline424.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="165" height="53" alt="int_0^infty(I_1(2sqrt(u))-sqrt(u))/((e^u-1)sqrt(u))du" /></td><td align="right" width="10"> <div id="eqn134" class="eqnum"> (134) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline425.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline426.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline427.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="159" height="44" alt="int_0^infty(_0F^~_1(;2;u)-1)/(e^u-1)du" /></td><td align="right" width="10"> <div id="eqn135" class="eqnum"> (135) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline428.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline429.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt=" approx " /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline430.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="64" height="20" alt="1.078189" /></td><td align="right" width="10"> <div id="eqn136" class="eqnum"> (136) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline431.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="18" height="20" alt="C_2" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline432.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline433.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="72" height="49" alt="sum_(n=1)^(infty)(zeta(2n))/(n!)" /></td><td align="right" width="10"> <div id="eqn137" class="eqnum"> (137) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline434.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline435.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline436.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="81" height="49" alt="sum_(n=1)^(infty)e^(1/n^2)-1" /></td><td align="right" width="10"> <div id="eqn138" class="eqnum"> (138) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline437.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline438.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline439.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="185" height="49" alt="int_0^1(u_0F_2(;3/2,2;1/4u^4))/(e^u-1)du" /></td><td align="right" width="10"> <div id="eqn139" class="eqnum"> (139) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline440.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline441.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt=" approx " /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline442.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="64" height="20" alt="2.407447" /></td><td align="right" width="10"> <div id="eqn140" class="eqnum"> (140) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline443.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="18" height="20" alt="C_3" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline444.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline445.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="72" height="49" alt="sum_(n=1)^(infty)(zeta(2n))/((2n)!)" /></td><td align="right" width="10"> <div id="eqn141" class="eqnum"> (141) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline446.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline447.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline448.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="264" height="45" alt="int_0^infty(u+_0F^~_1(;2;-u)-_0F^~_1(;2;u))/(2(1-e^u))du" /></td><td align="right" width="10"> <div id="eqn142" class="eqnum"> (142) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/RiemannZetaFunction/Inline449.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/RiemannZetaFunction/Inline450.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt=" approx " /></td><td align="left"><img src="/images/equations/RiemannZetaFunction/Inline451.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="68" height="20" alt="0.869002." /></td><td align="right" width="10"> <div id="eqn143" class="eqnum"> (143) </div> </td></tr> </table> </div> <p> (OEIS <a href="http://oeis.org/A093720">A093720</a>, <a href="http://oeis.org/A076813">A076813</a>, and <a href="http://oeis.org/A093721">A093721</a>), where <img src="/images/equations/RiemannZetaFunction/Inline452.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="36" height="21" alt="I_n(z)" /> is a <a href="/ModifiedBesselFunctionoftheFirstKind.html">modified Bessel function of the first kind</a>, <img src="/images/equations/RiemannZetaFunction/Inline453.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="24" alt="_pF^~_q" /> is a <a href="/RegularizedHypergeometricFunction.html">regularized hypergeometric function</a>. These sums have no known <a href="/Closed-FormSolution.html">closed-form</a> expression. </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="365.152" height="225.561" src="images/eps-svg/RiemannZetaInv_1000.svg" class="" alt="RiemannZetaInv" /> </div> <p> The inverse of the Riemann zeta function <img src="/images/equations/RiemannZetaFunction/Inline454.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="53" height="21" alt="1/zeta(p)" />, plotted above, is the asymptotic density of <img src="/images/equations/RiemannZetaFunction/Inline455.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="11" height="21" alt="p" />th-powerfree numbers (i.e., <a href="/Squarefree.html">squarefree</a> numbers, <a href="/Cubefree.html">cubefree</a> numbers, etc.). The following table gives the number <img src="/images/equations/RiemannZetaFunction/Inline456.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="45" height="24" alt="Q_p(n)" /> of <img src="/images/equations/RiemannZetaFunction/Inline457.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="11" height="21" alt="p" />th-powerfree numbers <img src="/images/equations/RiemannZetaFunction/Inline458.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="28" height="21" alt="<=n" /> for several values of <img src="/images/equations/RiemannZetaFunction/Inline459.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />. </p> <div class="table-responsive"> <table align="center" class="mathworldtable"> <tr style=""><td align="right"><img src="/images/equations/RiemannZetaFunction/Inline460.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="11" height="21" alt="p" /></td><td align="right"><img src="/images/equations/RiemannZetaFunction/Inline461.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="53" height="21" alt="1/zeta(p)" /></td><td align="right"><img src="/images/equations/RiemannZetaFunction/Inline462.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="24" alt="Q_p(10)" /></td><td align="right"><img src="/images/equations/RiemannZetaFunction/Inline463.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="63" height="24" alt="Q_p(100)" /></td><td align="right"><img src="/images/equations/RiemannZetaFunction/Inline464.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="61" height="25" alt="Q_p(10^3)" /></td><td align="right"><img src="/images/equations/RiemannZetaFunction/Inline465.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="61" height="25" alt="Q_p(10^4)" /></td><td align="right"><img src="/images/equations/RiemannZetaFunction/Inline466.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="61" height="25" alt="Q_p(10^5)" /></td><td align="right"><img src="/images/equations/RiemannZetaFunction/Inline467.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="61" height="25" alt="Q_p(10^6)" /></td></tr><tr style=""><td align="right">2</td><td align="right">0.607927</td><td align="right">7</td><td align="right">61</td><td align="right">608</td><td align="right">6083</td><td align="right">60794</td><td align="right">607926</td></tr><tr style=""><td align="right">3</td><td align="right">0.831907</td><td align="right">9</td><td align="right">85</td><td align="right">833</td><td align="right">8319</td><td align="right">83190</td><td align="right">831910</td></tr><tr style=""><td align="right">4</td><td align="right">0.923938</td><td align="right">10</td><td align="right">93</td><td align="right">925</td><td align="right">9240</td><td align="right">92395</td><td align="right">923939</td></tr><tr style=""><td align="right">5</td><td align="right">0.964387</td><td align="right">10</td><td align="right">97</td><td align="right">965</td><td align="right">9645</td><td align="right">96440</td><td align="right">964388</td></tr><tr style=""><td align="right">6</td><td align="right">0.982953</td><td align="right">10</td><td align="right">99</td><td align="right">984</td><td align="right">9831</td><td align="right">98297</td><td align="right">982954</td></tr> </table> </div> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/AbelsFunctionalEquation.html">Abel's Functional Equation</a>, <a href="/BerryConjecture.html">Berry Conjecture</a>, <a href="/CriticalLine.html">Critical Line</a>, <a href="/CriticalStrip.html">Critical Strip</a>, <a href="/DebyeFunctions.html">Debye Functions</a>, <a href="/DirichletBetaFunction.html">Dirichlet Beta Function</a>, <a href="/DirichletEtaFunction.html">Dirichlet Eta Function</a>, <a href="/DirichletLambdaFunction.html">Dirichlet Lambda Function</a>, <a href="/EulerProduct.html">Euler Product</a>, <a href="/HarmonicSeries.html">Harmonic Series</a>, <a href="/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>, <a href="/KhinchinsConstant.html">Khinchin's Constant</a>, <a href="/LehmersPhenomenon.html">Lehmer's Phenomenon</a>, <a href="/MontgomerysPairCorrelationConjecture.html">Montgomery's Pair Correlation Conjecture</a>, <a href="/p-Series.html"><i>p</i>-Series</a>, <a href="/PeriodicZetaFunction.html">Periodic Zeta Function</a>, <a href="/PrimeNumberTheorem.html">Prime Number Theorem</a>, <a href="/PsiFunction.html">Psi Function</a>, <a href="/RiemannHypothesis.html">Riemann Hypothesis</a>, <a href="/RiemannP-Series.html">Riemann P-Series</a>, <a href="/Riemann-SiegelFunctions.html">Riemann-Siegel Functions</a>, <a href="/Riemann-vonMangoldtFormula.html">Riemann-von Mangoldt Formula</a>, <a href="/RiemannZetaFunctionZeta2.html">Riemann Zeta Function <i>zeta</i>(2)</a>, <a href="/RiemannZetaFunctionZeros.html">Riemann Zeta Function Zeros</a>, <a href="/StieltjesConstants.html">Stieltjes Constants</a>, <a href="/VoroninUniversalityTheorem.html">Voronin Universality Theorem</a>, <a href="/Xi-Function.html">Xi-Function</a> <a href="/classroom/RiemannZetaFunction.html" class="explore-classroom">Explore this topic in the MathWorld classroom</a>   <h2>Related Wolfram sites</h2><a href="http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/">http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/</a>   <p class="contributor"> <i>Portions of this entry contributed by <a target="_blank" href="/topics/Sondow.html">Jonathan Sondow</a> (<a target="_blank" href="http://home.earthlink.net/~jsondow/">author's link</a>)</i> </p>   <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=Riemann+hypothesis"> Riemann hypothesis </a> </li> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=riemann+zeta+function+series+representation"> riemann zeta function series representation </a> </li> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=riemann+zeta+function%28a%2Bb*i%29"> riemann zeta function(a+b*i) </a> </li> </ul> </div> </div>   <h2>References</h2><cite>Abramowitz, M. and Stegun, I. A. 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"Searching Symbolically for Apéry-Like Formulae for Values of the Riemann Zeta Function." <i>ACM SIGSAM Bull. Algebraic Sym. Manip.</i> <b>30</b>, 2-7, 1996.</cite><cite>Borwein, J. M. and Bradley, D. M. "Empirically Determined Apéry-Like Formulae for <img src="/images/equations/RiemannZetaFunction/Inline475.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="72" height="21" alt="zeta(4n+3)" />." <i>Exp. Math.</i> <b>6</b>, 181-194, 1997.</cite><cite>Borwein, J. M.; Bradley, D. M.; and Crandall, R. E. "Computational Strategies for the Riemann Zeta Function." <i>J. Comput. Appl. Math.</i> <b>121</b>, 247-296, 2000.</cite><cite>Castellanos, D. "The Ubiquitous Pi. Part I." <i>Math. Mag.</i> <b>61</b>, 67-98, 1988.</cite><cite>Choudhury, B. K. "The Riemann Zeta-Function and Its Derivatives." <i>Proc. Roy. Soc. London Ser. A</i> <b>450</b>, 477-499, 1995.</cite><cite>Cohen, H. "High Precision Computation of Hardy-Littlewood Constants." 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Sequences <a href="http://oeis.org/A001067">A001067</a>, <a href="http://oeis.org/A002432">A002432</a>/M4283, <a href="http://oeis.org/A006953">A006953</a>/M2039, <a href="http://oeis.org/A057866">A057866</a>, <a href="http://oeis.org/A057867">A057867</a>, <a href="http://oeis.org/A059750">A059750</a>, <a href="http://oeis.org/A073002">A073002</a>, <a href="http://oeis.org/A076813">A076813</a>, <a href="http://oeis.org/A093720">A093720</a>, <a href="http://oeis.org/A093721">A093721</a>, <a href="http://oeis.org/A114474">A114474</a>, <a href="http://oeis.org/A114875">A114875</a>, <a href="http://oeis.org/A117972">A117972</a>, and <a href="http://oeis.org/A117973">A117973</a> in "The On-Line Encyclopedia of Integer Sequences."</cite><cite>Sondow, J. "Analytic Continuation of Riemann's Zeta Function and Values at Negative Integers via Euler's Transformation of Series." <i>Proc. Amer. Math. Soc.</i> <b>120</b>, 421-424, 1994.</cite><cite>Spanier, J. and Oldham, K. B. "The Zeta Numbers and Related Functions." Ch. 3 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387488065/ref=nosim/ericstreasuretro">An Atlas of Functions.</a></i> Washington, DC: Hemisphere, pp. 25-33, 1987.</cite><cite>Srivastava, H. M. "Some Simple Algorithms for the Evaluations and Representations of the Riemann Zeta Function at Positive Integer Arguments." <i>J. Math. Anal. Appl.</i> <b>246</b>, 331-351, 2000.</cite><cite>Stark, E. L. "The Series <img src="/images/equations/RiemannZetaFunction/Inline487.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="60" height="23" alt="sum_(k=1)^(infty)k^(-s)" /> <img src="/images/equations/RiemannZetaFunction/Inline488.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="s=2" />, 3, 4, ..., Once More." <i>Math. Mag.</i> <b>47</b>, 197-202, 1974.</cite><cite>Stieltjes, T. J. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387555609/ref=nosim/ericstreasuretro">Oeuvres Complètes, Vol. 2</a></i> (Ed. G. van Dijk.) New York: Springer-Verlag, p. 100, 1993.</cite><cite>Titchmarsh, E. C. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0198533691/ref=nosim/ericstreasuretro">The Zeta-Function of Riemann.</a></i> London: Cambridge University Press, 1930.</cite><cite>Titchmarsh, E. C. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0198533691/ref=nosim/ericstreasuretro">The Theory of the Riemann Zeta Function, 2nd ed.</a></i> New York: Clarendon Press, 1987.</cite><cite>Tyler, D. and Chernhoff, P. "Problem 3103. An Odd Sum Reappears." <i>Amer. Math. Monthly</i> <b>92</b>, 507, 1985.</cite><cite>Vardi, I. "The Riemann Zeta Function." Ch. 8 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0685479412/ref=nosim/ericstreasuretro">Computational Recreations in Mathematica.</a></i> Reading, MA: Addison-Wesley, pp. 141-174, 1991.</cite><cite>Wagon, S. "The Riemann Zeta Function." §10.6 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387753664/ref=nosim/ericstreasuretro">Mathematica in Action.</a></i> New York: W. H. Freeman, pp. 353-362, 1991.</cite><cite>Watkins,M. R. "Inexplicable Secrets of Creation." <a href="http://www.maths.ex.ac.uk/~mwatkins/zeta/">http://www.maths.ex.ac.uk/~mwatkins/zeta/</a>.</cite><cite>Weisstein, E. W. "Books about Riemann Zeta Function." <a href="http://www.ericweisstein.com/encyclopedias/books/RiemannZetaFunction.html">http://www.ericweisstein.com/encyclopedias/books/RiemannZetaFunction.html</a>.</cite><cite>Wells, D. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0140080295/ref=nosim/ericstreasuretro">The Penguin Dictionary of Curious and Interesting Numbers.</a></i> Middlesex, England: Penguin Books, 1986.</cite><cite>Whittaker, E. T. and Watson, G. N. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521091896/ref=nosim/ericstreasuretro">A Course in Modern Analysis, 4th ed.</a></i> Cambridge, England: Cambridge University Press, 1990.</cite><cite>Woon, S. C. "Generalization of a Relation Between the Riemann Zeta Function and Bernoulli Numbers." 24 Dec 1998. <a href="http://arxiv.org/abs/math.NT/9812143">http://arxiv.org/abs/math.NT/9812143</a>.</cite><cite>Zucker, I. J. "The Summation of Series of Hyperbolic Functions." <i>SIAM J. Math. Anal.</i> <b>10</b>, 192-206, 1979.</cite><cite>Zucker, I. J. "Some Infinite Series of Exponential and Hyperbolic Functions." <i>SIAM J. Math. Anal.</i> <b>15</b>, 406-413, 1984.</cite><cite>Zudilin, W. "One of the Numbers <img src="/images/equations/RiemannZetaFunction/Inline489.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(5)" />, <img src="/images/equations/RiemannZetaFunction/Inline490.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(7)" />, <img src="/images/equations/RiemannZetaFunction/Inline491.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="zeta(9)" />, <img src="/images/equations/RiemannZetaFunction/Inline492.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="40" height="21" alt="zeta(11)" /> Is Irrational." <i>Uspekhi Mat. Nauk</i> <b>56</b>, 149-150, 2001.</cite><cite>Zvengrowski, P. and Saidak, F. "On the Modulus of the Riemann Zeta Function in the Critical Strip." <i>Math. Slovaca</i> <b>53</b>, 145-172, 2003.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/riemann_zeta_function/kl/w4/u3/" title="Riemann Zeta Function" target="_blank">Riemann Zeta Function</a>   <h2>Cite this as:</h2> <p> <a href="/topics/Sondow.html">Sondow, Jonathan</a> and <a href="/about/author.html">Weisstein, Eric W.</a> "Riemann Zeta Function." 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