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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>grand Riemann hypothesis</b> is a generalisation of the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> and <a href="/wiki/Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">generalized Riemann hypothesis</a>. It states that the nontrivial zeros of all <a href="/wiki/Automorphic_L-function" title="Automorphic L-function">automorphic <i>L</i>-functions</a> lie on the critical line <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}+it}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}+it}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/652fd6b60d36ab5d3196b78d37714c0e272d5ac7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.481ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}+it}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> a real number variable and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>. </p><p>The <b>modified grand Riemann hypothesis</b> is the assertion that the nontrivial zeros of all automorphic <i>L</i>-functions lie on the critical line or the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grand_Riemann_hypothesis&action=edit&section=1" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Robert_Langlands" title="Robert Langlands">Robert Langlands</a>, in his general <a href="/wiki/Langlands_program#Functoriality" title="Langlands program">functoriality conjectures</a>, asserts that all global <i>L</i>-functions should be automorphic.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="/wiki/Siegel_zero" title="Siegel zero">Siegel zero</a>, conjectured not to exist,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> is a possible real zero of a <a href="/wiki/Dirichlet_L-series" class="mw-redirect" title="Dirichlet L-series">Dirichlet <i>L</i>-series</a>, rather near <i>s</i> = 1.</li> <li><i>L</i>-functions of Maass cusp forms can have trivial zeros which are off the real line.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grand_Riemann_hypothesis&action=edit&section=2" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSarnak2005" class="citation book cs1"><a href="/wiki/Peter_Sarnak" title="Peter Sarnak">Sarnak, Peter</a> (2005). <a rel="nofollow" class="external text" href="http://web.math.princeton.edu/sarnak/FieldNotesCurrent.pdf">"Notes on the Generalized Ramanujan Conjectures"</a> <span class="cs1-format">(PDF)</span>. In <a href="/wiki/James_Arthur_(mathematician)" title="James Arthur (mathematician)">Arthur, James</a>; Ellwood, David; <a href="/wiki/Robert_Kottwitz" title="Robert Kottwitz">Kottwitz, Robert</a> (eds.). <i>Harmonic Analysis, The Trace Formula, and Shimura Varieties</i>. Vol. 4. Princeton: <a href="/wiki/Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a>. Clay Mathematics Proceedings. pp. 659–685. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-3844-X" title="Special:BookSources/0-8218-3844-X"><bdi>0-8218-3844-X</bdi></a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1534-6455">1534-6455</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/637721920">637721920</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151004063221/http://web.math.princeton.edu/sarnak/FieldNotesCurrent.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on October 4, 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">November 11,</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Notes+on+the+Generalized+Ramanujan+Conjectures&rft.btitle=Harmonic+Analysis%2C+The+Trace+Formula%2C+and+Shimura+Varieties&rft.place=Princeton&rft.pages=659-685&rft.pub=Clay+Mathematics+Institute.+Clay+Mathematics+Proceedings&rft.date=2005&rft_id=info%3Aoclcnum%2F637721920&rft.issn=1534-6455&rft.isbn=0-8218-3844-X&rft.aulast=Sarnak&rft.aufirst=Peter&rft_id=http%3A%2F%2Fweb.math.princeton.edu%2Fsarnak%2FFieldNotesCurrent.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrand+Riemann+hypothesis" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConreyIwaniec2002" class="citation journal cs1"><a href="/wiki/Brian_Conrey" title="Brian Conrey">Conrey, Brian</a>; <a href="/wiki/Henryk_Iwaniec" title="Henryk Iwaniec">Iwaniec, Henryk</a> (2002). <a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Faa103-3-5">"Spacing of zeros of Hecke L-functions and the class number problem"</a>. <i><a href="/wiki/Acta_Arithmetica" title="Acta Arithmetica">Acta Arithmetica</a></i>. <b>103</b> (3): 259–312. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0111012">math/0111012</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002AcAri.103..259C">2002AcAri.103..259C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Faa103-3-5">10.4064/aa103-3-5</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0065-1036">0065-1036</a>. <q>Conrey and Iwaniec show that sufficiently many small gaps between zeros of the Riemann zeta function would imply the non-existence of Landau–Siegel zeros.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Arithmetica&rft.atitle=Spacing+of+zeros+of+Hecke+L-functions+and+the+class+number+problem&rft.volume=103&rft.issue=3&rft.pages=259-312&rft.date=2002&rft_id=info%3Aarxiv%2Fmath%2F0111012&rft.issn=0065-1036&rft_id=info%3Adoi%2F10.4064%2Faa103-3-5&rft_id=info%3Abibcode%2F2002AcAri.103..259C&rft.aulast=Conrey&rft.aufirst=Brian&rft.au=Iwaniec%2C+Henryk&rft_id=https%3A%2F%2Fdoi.org%2F10.4064%252Faa103-3-5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrand+Riemann+hypothesis" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grand_Riemann_hypothesis&action=edit&section=3" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Borwein, Peter B. (2008), <i>The Riemann hypothesis: a resource for the aficionado and virtuoso alike</i>, CMS books in mathematics, vol. 27, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-72125-5" title="Special:BookSources/978-0-387-72125-5"><bdi>978-0-387-72125-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Riemann+hypothesis%3A+a+resource+for+the+aficionado+and+virtuoso+alike&rft.series=CMS+books+in+mathematics&rft.pub=Springer-Verlag&rft.date=2008&rft.isbn=978-0-387-72125-5&rft.aulast=Borwein&rft.aufirst=Peter+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrand+Riemann+hypothesis" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output 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