Riemann zeta function

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href="#Riemann's_functional_equation"> <div class="vector-toc-text"> 3 Riemann's functional equation </div> </a> <button aria-controls="toc-Riemann's_functional_equation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Riemann's functional equation subsection </button> <ul id="toc-Riemann's_functional_equation-sublist" class="vector-toc-list"> <li id="toc-Equivalencies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalencies"> <div class="vector-toc-text"> 3.1 Equivalencies </div> </a> <ul id="toc-Equivalencies-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Zeros,_the_critical_line,_and_the_Riemann_hypothesis" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Zeros,_the_critical_line,_and_the_Riemann_hypothesis"> <div class="vector-toc-text"> 4 Zeros, the critical line, and the Riemann hypothesis </div> </a> <button aria-controls="toc-Zeros,_the_critical_line,_and_the_Riemann_hypothesis-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Zeros, the critical line, and the Riemann hypothesis subsection </button> <ul id="toc-Zeros,_the_critical_line,_and_the_Riemann_hypothesis-sublist" class="vector-toc-list"> <li id="toc-Number_of_zeros_in_the_critical_strip" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Number_of_zeros_in_the_critical_strip"> <div class="vector-toc-text"> 4.1 Number of zeros in the critical strip </div> </a> <ul id="toc-Number_of_zeros_in_the_critical_strip-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Hardy–Littlewood_conjectures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Hardy–Littlewood_conjectures"> <div class="vector-toc-text"> 4.2 The Hardy–Littlewood conjectures </div> </a> <ul id="toc-The_Hardy–Littlewood_conjectures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zero-free_region" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zero-free_region"> <div class="vector-toc-text"> 4.3 Zero-free region </div> </a> <ul id="toc-Zero-free_region-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_results" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_results"> <div class="vector-toc-text"> 4.4 Other results </div> </a> <ul id="toc-Other_results-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Specific_values" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Specific_values"> <div class="vector-toc-text"> 5 Specific values </div> </a> <ul id="toc-Specific_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Various_properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Various_properties"> <div class="vector-toc-text"> 6 Various properties </div> </a> <button aria-controls="toc-Various_properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Various properties subsection </button> <ul id="toc-Various_properties-sublist" class="vector-toc-list"> <li id="toc-Reciprocal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reciprocal"> <div class="vector-toc-text"> 6.1 Reciprocal </div> </a> <ul id="toc-Reciprocal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Universality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Universality"> <div class="vector-toc-text"> 6.2 Universality </div> </a> <ul id="toc-Universality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Estimates_of_the_maximum_of_the_modulus_of_the_zeta_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estimates_of_the_maximum_of_the_modulus_of_the_zeta_function"> <div class="vector-toc-text"> 6.3 Estimates of the maximum of the modulus of the zeta function </div> </a> <ul id="toc-Estimates_of_the_maximum_of_the_modulus_of_the_zeta_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_argument_of_the_Riemann_zeta_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_argument_of_the_Riemann_zeta_function"> <div class="vector-toc-text"> 6.4 The argument of the Riemann zeta function </div> </a> <ul id="toc-The_argument_of_the_Riemann_zeta_function-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Representations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Representations"> <div class="vector-toc-text"> 7 Representations </div> </a> <button aria-controls="toc-Representations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Representations subsection </button> <ul id="toc-Representations-sublist" class="vector-toc-list"> <li id="toc-Dirichlet_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dirichlet_series"> <div class="vector-toc-text"> 7.1 Dirichlet series </div> </a> <ul id="toc-Dirichlet_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mellin-type_integrals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mellin-type_integrals"> <div class="vector-toc-text"> 7.2 Mellin-type integrals </div> </a> <ul id="toc-Mellin-type_integrals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theta_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theta_functions"> <div class="vector-toc-text"> 7.3 Theta functions </div> </a> <ul id="toc-Theta_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Laurent_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Laurent_series"> <div class="vector-toc-text"> 7.4 Laurent series </div> </a> <ul id="toc-Laurent_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integral" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral"> <div class="vector-toc-text"> 7.5 Integral </div> </a> <ul id="toc-Integral-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rising_factorial" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rising_factorial"> <div class="vector-toc-text"> 7.6 Rising factorial </div> </a> <ul id="toc-Rising_factorial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hadamard_product" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hadamard_product"> <div class="vector-toc-text"> 7.7 Hadamard product </div> </a> <ul id="toc-Hadamard_product-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Globally_convergent_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Globally_convergent_series"> <div class="vector-toc-text"> 7.8 Globally convergent series </div> </a> <ul id="toc-Globally_convergent_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rapidly_convergent_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rapidly_convergent_series"> <div class="vector-toc-text"> 7.9 Rapidly convergent series </div> </a> <ul id="toc-Rapidly_convergent_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Series_representation_at_positive_integers_via_the_primorial" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Series_representation_at_positive_integers_via_the_primorial"> <div class="vector-toc-text"> 7.10 Series representation at positive integers via the primorial </div> </a> <ul id="toc-Series_representation_at_positive_integers_via_the_primorial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Series_representation_by_the_incomplete_poly-Bernoulli_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Series_representation_by_the_incomplete_poly-Bernoulli_numbers"> <div class="vector-toc-text"> 7.11 Series representation by the incomplete poly-Bernoulli numbers </div> </a> <ul id="toc-Series_representation_by_the_incomplete_poly-Bernoulli_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Mellin_transform_of_the_Engel_map" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Mellin_transform_of_the_Engel_map"> <div class="vector-toc-text"> 7.12 The Mellin transform of the Engel map </div> </a> <ul id="toc-The_Mellin_transform_of_the_Engel_map-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Thue-Morse_sequence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Thue-Morse_sequence"> <div class="vector-toc-text"> 7.13 Thue-Morse sequence </div> </a> <ul id="toc-Thue-Morse_sequence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_nth_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_nth_dimensions"> <div class="vector-toc-text"> 7.14 In nth dimensions </div> </a> <ul id="toc-In_nth_dimensions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Numerical_algorithms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Numerical_algorithms"> <div class="vector-toc-text"> 8 Numerical algorithms </div> </a> <ul id="toc-Numerical_algorithms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 9 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Musical_tuning" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Musical_tuning"> <div class="vector-toc-text"> 9.1 Musical tuning </div> </a> <ul id="toc-Musical_tuning-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinite_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinite_series"> <div class="vector-toc-text"> 9.2 Infinite series </div> </a> <ul id="toc-Infinite_series-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 10 Generalizations </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 11 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 12 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> 13 Sources </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 14 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Riemann zeta function</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 54 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-54" aria-hidden="true" > 54 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D9%84%D8%A9_%D8%B2%D9%8A%D8%AA%D8%A7_%D9%84%D8%B1%D9%8A%D9%85%D8%A7%D9%86" title="دالة زيتا لريمان – Arabic" lang="ar" hreflang="ar" data-title="دالة زيتا لريمان" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Rieman_zeta_funksiyas%C4%B1" title="Rieman zeta funksiyası – Azerbaijani" lang="az" hreflang="az" data-title="Rieman zeta funksiyası" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9C%E0%A7%87%E0%A6%9F%E0%A6%BE_%E0%A6%AB%E0%A6%BE%E0%A6%82%E0%A6%B6%E0%A6%A8" title="জেটা ফাংশন – Bangla" lang="bn" hreflang="bn" data-title="জেটা ফাংশন" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D0%B7%D1%8D%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%8F_%D0%A0%D1%8B%D0%BC%D0%B0%D0%BD%D0%B0" title="Дзэта-функцыя Рымана – Belarusian" lang="be" hreflang="be" data-title="Дзэта-функцыя Рымана" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B7%D0%B5%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD" title="Дзета-функция на Риман – Bulgarian" lang="bg" hreflang="bg" data-title="Дзета-функция на Риман" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Riemannova_zeta-funkcija" title="Riemannova zeta-funkcija – Bosnian" lang="bs" hreflang="bs" data-title="Riemannova zeta-funkcija" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_zeta_de_Riemann" title="Funció zeta de Riemann – Catalan" lang="ca" hreflang="ca" data-title="Funció zeta de Riemann" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Riemannova_funkce_zeta" title="Riemannova funkce zeta – Czech" lang="cs" hreflang="cs" data-title="Riemannova funkce zeta" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Riemanns_zetafunktion" title="Riemanns zetafunktion – Danish" lang="da" hreflang="da" data-title="Riemanns zetafunktion" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://de.wikipedia.org/wiki/Riemannsche_Zeta-Funktion" title="Riemannsche Zeta-Funktion – German" lang="de" hreflang="de" data-title="Riemannsche Zeta-Funktion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Riemanni_dzeetafunktsioon" title="Riemanni dzeetafunktsioon – Estonian" lang="et" hreflang="et" data-title="Riemanni dzeetafunktsioon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7_%CE%B6%CE%AE%CF%84%CE%B1_%CE%A1%CE%AF%CE%BC%CE%B1%CE%BD" title="Συνάρτηση ζήτα Ρίμαν – Greek" lang="el" hreflang="el" data-title="Συνάρτηση ζήτα Ρίμαν" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_zeta_de_Riemann" title="Función zeta de Riemann – Spanish" lang="es" hreflang="es" data-title="Función zeta de Riemann" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Rimana_%CE%B6_funkcio" title="Rimana ζ funkcio – Esperanto" lang="eo" hreflang="eo" data-title="Rimana ζ funkcio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_%D8%B2%D8%AA%D8%A7%DB%8C_%D8%B1%DB%8C%D9%85%D8%A7%D9%86" title="تابع زتای ریمان – Persian" lang="fa" hreflang="fa" data-title="تابع زتای ریمان" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fonction_z%C3%AAta_de_Riemann" title="Fonction zêta de Riemann – French" lang="fr" hreflang="fr" data-title="Fonction zêta de Riemann" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n_zeta_de_Riemann" title="Función zeta de Riemann – Galician" lang="gl" hreflang="gl" data-title="Función zeta de Riemann" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A6%AC%EB%A7%8C_%EC%A0%9C%ED%83%80_%ED%95%A8%EC%88%98" title="리만 제타 함수 – Korean" lang="ko" hreflang="ko" data-title="리만 제타 함수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8C%D5%AB%D5%B4%D5%A1%D5%B6%D5%AB_%D5%A6%D5%A5%D5%BF%D5%A1_%D6%86%D5%B8%D6%82%D5%B6%D5%AF%D6%81%D5%AB%D5%A1" title="Ռիմանի զետա ֆունկցիա – Armenian" lang="hy" hreflang="hy" data-title="Ռիմանի զետա ֆունկցիա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B0%E0%A5%80%E0%A4%AE%E0%A4%BE%E0%A4%A8_%E0%A4%9C%E0%A5%80%E0%A4%9F%E0%A4%BE_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="रीमान जीटा फलन – Hindi" lang="hi" hreflang="hi" data-title="रीमान जीटा फलन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Riemannova_zeta-funkcija" title="Riemannova zeta-funkcija – Croatian" lang="hr" hreflang="hr" data-title="Riemannova zeta-funkcija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_zeta_Riemann" title="Fungsi zeta Riemann – Indonesian" lang="id" hreflang="id" data-title="Fungsi zeta Riemann" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Zetufall_Riemanns" title="Zetufall Riemanns – Icelandic" lang="is" hreflang="is" data-title="Zetufall Riemanns" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_zeta_di_Riemann" title="Funzione zeta di Riemann – Italian" lang="it" hreflang="it" data-title="Funzione zeta di Riemann" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%99%D7%AA_%D7%96%D7%98%D7%90_%D7%A9%D7%9C_%D7%A8%D7%99%D7%9E%D7%9F" title="פונקציית זטא של רימן – Hebrew" lang="he" hreflang="he" data-title="פונקציית זטא של רימן" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Fonksyon_zeta_Riemann" title="Fonksyon zeta Riemann – Haitian Creole" lang="ht" hreflang="ht" data-title="Fonksyon zeta Riemann" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Functio_zeta_Riemanniana" title="Functio zeta Riemanniana – Latin" lang="la" hreflang="la" data-title="Functio zeta Riemanniana" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Rymano_dzeta_funkcija" title="Rymano dzeta funkcija – Lithuanian" lang="lt" hreflang="lt" data-title="Rymano dzeta funkcija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Riemann-f%C3%A9le_z%C3%A9ta-f%C3%BCggv%C3%A9ny" title="Riemann-féle zéta-függvény – Hungarian" lang="hu" hreflang="hu" data-title="Riemann-féle zéta-függvény" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Riemann-z%C3%A8ta-functie" title="Riemann-zèta-functie – Dutch" lang="nl" hreflang="nl" data-title="Riemann-zèta-functie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%AA%E3%83%BC%E3%83%9E%E3%83%B3%E3%82%BC%E3%83%BC%E3%82%BF%E9%96%A2%E6%95%B0" title="リーマンゼータ関数 – Japanese" lang="ja" hreflang="ja" data-title="リーマンゼータ関数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Riemanns_zetafunksjon" title="Riemanns zetafunksjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Riemanns zetafunksjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Riemann_zeta_funksiyasi" title="Riemann zeta funksiyasi – Uzbek" lang="uz" hreflang="uz" data-title="Riemann zeta funksiyasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja_dzeta_Riemanna" title="Funkcja dzeta Riemanna – Polish" lang="pl" hreflang="pl" data-title="Funkcja dzeta Riemanna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_zeta_de_Riemann" title="Função zeta de Riemann – Portuguese" lang="pt" hreflang="pt" data-title="Função zeta de Riemann" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Func%C8%9Bia_zeta_Riemann" title="Funcția zeta Riemann – Romanian" lang="ro" hreflang="ro" data-title="Funcția zeta Riemann" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B7%D0%B5%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD%D0%B0" title="Дзета-функция Римана – Russian" lang="ru" hreflang="ru" data-title="Дзета-функция Римана" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Zeta_di_Riemann" title="Zeta di Riemann – Sicilian" lang="scn" hreflang="scn" data-title="Zeta di Riemann" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Riemann_zeta_function" title="Riemann zeta function – Simple English" lang="en-simple" hreflang="en-simple" data-title="Riemann zeta function" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Riemannova_zeta_funkcia" title="Riemannova zeta funkcia – Slovak" lang="sk" hreflang="sk" data-title="Riemannova zeta funkcia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Riemannova_funkcija_zeta" title="Riemannova funkcija zeta – Slovenian" lang="sl" hreflang="sl" data-title="Riemannova funkcija zeta" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD%D0%BE%D0%B2%D0%B0_%D0%B7%D0%B5%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Риманова зета-функција – Serbian" lang="sr" hreflang="sr" data-title="Риманова зета-функција" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Riemannova_zeta-funkcija" title="Riemannova zeta-funkcija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Riemannova zeta-funkcija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Riemannin_zeeta-funktio" title="Riemannin zeeta-funktio – Finnish" lang="fi" hreflang="fi" data-title="Riemannin zeeta-funktio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Riemanns_zetafunktion" title="Riemanns zetafunktion – Swedish" lang="sv" hreflang="sv" data-title="Riemanns zetafunktion" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B0%E0%AF%80%E0%AE%AE%E0%AE%A9%E0%AF%8D_%E0%AE%87%E0%AE%9A%E0%AF%80%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%BE_%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AE%BF%E0%AE%AF%E0%AE%AE%E0%AF%8D" title="ரீமன் இசீட்டா சார்பியம் – Tamil" lang="ta" hreflang="ta" data-title="ரீமன் இசீட்டா சார்பியம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD_%D0%B7%D0%B5%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F%D1%81%D0%B5" title="Риман зета-функциясе – Tatar" lang="tt" hreflang="tt" data-title="Риман зета-функциясе" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99%E0%B8%8B%E0%B8%B5%E0%B8%95%E0%B8%B2%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%A3%E0%B8%B5%E0%B8%A1%E0%B8%B1%E0%B8%99" title="ฟังก์ชันซีตาของรีมัน – Thai" lang="th" hreflang="th" data-title="ฟังก์ชันซีตาของรีมัน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Riemann_zeta_i%C5%9Flevi" title="Riemann zeta işlevi – Turkish" lang="tr" hreflang="tr" data-title="Riemann zeta işlevi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B7%D0%B5%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%A0%D1%96%D0%BC%D0%B0%D0%BD%D0%B0" title="Дзета-функція Рімана – Ukrainian" lang="uk" hreflang="uk" data-title="Дзета-функція Рімана" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B1%DB%8C%D9%85%D8%A7%D9%86_%D8%B2%DB%8C%D9%B9%D8%A7_%D9%81%D9%86%DA%A9%D8%B4%D9%86" title="ریمان زیٹا فنکشن – Urdu" lang="ur" hreflang="ur" data-title="ریمان زیٹا فنکشن" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%A0m_zeta_Riemann" title="Hàm zeta Riemann – Vietnamese" lang="vi" hreflang="vi" data-title="Hàm zeta Riemann" data-language-autonym="Tiếng Việt" 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decoding="async" width="250" height="231" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Cplot_zeta.svg/375px-Cplot_zeta.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Cplot_zeta.svg/500px-Cplot_zeta.svg.png 2x" data-file-width="462" data-file-height="426" /></a><figcaption>The Riemann zeta function ζ(z) plotted with <a href="/wiki/Domain_coloring" title="Domain coloring">domain coloring</a>.<a href="#cite_note-1">[1]</a></figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Riemann-Zeta-Detail.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Riemann-Zeta-Detail.png/200px-Riemann-Zeta-Detail.png" decoding="async" width="200" height="462" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Riemann-Zeta-Detail.png/300px-Riemann-Zeta-Detail.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Riemann-Zeta-Detail.png/400px-Riemann-Zeta-Detail.png 2x" data-file-width="2080" data-file-height="4808" /></a><figcaption>The pole at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/078535cde78d90bfa1d9fbb2446204593a921d57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=1}"> and two zeros on the critical line.</figcaption></figure> The Riemann zeta function or Euler–Riemann zeta function, denoted by the <a href="/wiki/Greek_alphabet" title="Greek alphabet">Greek letter</a> ζ (<a href="/wiki/Zeta" title="Zeta">zeta</a>), is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">mathematical function</a> of a <a href="/wiki/Complex_variable" class="mw-redirect" title="Complex variable">complex variable</a> defined as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a171e24fc1c6d990c65e7fe82805613da4645645" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.418ex; height:6.843ex;" alt="{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots }"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e22215d90eb4893d05f9b2bff409094a3451f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>1}">, and its <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a> elsewhere.<a href="#cite_note-:0-2">[2]</a> The Riemann zeta function plays a pivotal role in <a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic number theory</a> and has applications in <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, and applied <a href="/wiki/Statistics" title="Statistics">statistics</a>. <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> first introduced and studied the function over the <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">reals</a> in the first half of the eighteenth century. <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a>'s 1859 article "<a href="/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude" title="On the Number of Primes Less Than a Given Magnitude">On the Number of Primes Less Than a Given Magnitude</a>" extended the Euler definition to a <a href="/wiki/Complex_number" title="Complex number">complex</a> variable, proved its <a href="/wiki/Meromorphic" class="mw-redirect" title="Meromorphic">meromorphic</a> continuation and <a href="/wiki/Functional_equation" title="Functional equation">functional equation</a>, and established a relation between its <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">zeros</a> and <a href="/wiki/Prime_number_theorem" title="Prime number theorem">the distribution of prime numbers</a>. This paper also contained the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>, a <a href="/wiki/Conjecture" title="Conjecture">conjecture</a> about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider the most important unsolved problem in <a href="/wiki/Pure_mathematics" title="Pure mathematics">pure mathematics</a>.<a href="#cite_note-3">[3]</a> The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ(2), provides a solution to the <a href="/wiki/Basel_problem" title="Basel problem">Basel problem</a>. In 1979 <a href="/wiki/Roger_Ap%C3%A9ry" title="Roger Apéry">Roger Apéry</a> proved the irrationality of <a href="/wiki/Ap%C3%A9ry%27s_constant" title="Apéry's constant">ζ(3)</a>. The values at negative integer points, also found by Euler, are <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> and play an important role in the theory of <a href="/wiki/Modular_form" title="Modular form">modular forms</a>. Many generalizations of the Riemann zeta function, such as <a href="/wiki/Dirichlet_series" title="Dirichlet series">Dirichlet series</a>, <a href="/wiki/Dirichlet_L-function" title="Dirichlet L-function">Dirichlet L-functions</a> and <a href="/wiki/L-function" title="L-function">L-functions</a>, are known. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ueber_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Gr%C3%B6sse.pdf" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Ueber_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Gr%C3%B6sse.pdf/page1-170px-Ueber_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Gr%C3%B6sse.pdf.jpg" decoding="async" width="170" height="284" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Ueber_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Gr%C3%B6sse.pdf/page1-255px-Ueber_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Gr%C3%B6sse.pdf.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Ueber_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Gr%C3%B6sse.pdf/page1-340px-Ueber_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Gr%C3%B6sse.pdf.jpg 2x" data-file-width="779" data-file-height="1300" /></a><figcaption>Bernhard Riemann's article On the number of primes below a given magnitude</figcaption></figure> The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it, where σ and t are real numbers. (The notation s, σ, and t is used traditionally in the study of the zeta function, following Riemann.) When Re(s) = σ > 1, the function can be written as a converging summation or as an integral: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\,\mathrm {d} x\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\,\mathrm {d} x\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97dc05324d2b62fe317eeb67a7a742091eed120e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.609ex; height:6.843ex;" alt="{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\,\mathrm {d} x\,,}"></dd></dl> where <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (s)=\int _{0}^{\infty }x^{s-1}\,e^{-x}\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (s)=\int _{0}^{\infty }x^{s-1}\,e^{-x}\,\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34b496dcab4417e3a8fdfb028115bae2fbad90fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.542ex; height:5.843ex;" alt="{\displaystyle \Gamma (s)=\int _{0}^{\infty }x^{s-1}\,e^{-x}\,\mathrm {d} x}"></dd></dl> is the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>. The Riemann zeta function is defined for other complex values via <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a> of the function defined for σ > 1. <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> considered the above series in 1740 for positive integer values of s, and later <a href="/wiki/Chebyshev" class="mw-redirect" title="Chebyshev">Chebyshev</a> extended the definition to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c05b7a57c920e79cf69fec018eeb50fa62627788" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.55ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>1.}"><a href="#cite_note-devlin-4">[4]</a> The above series is a prototypical <a href="/wiki/Dirichlet_series" title="Dirichlet series">Dirichlet series</a> that <a href="/wiki/Absolute_convergence" title="Absolute convergence">converges absolutely</a> to an <a href="/wiki/Analytic_function" title="Analytic function">analytic function</a> for s such that σ > 1 and <a href="/wiki/Divergent_series" title="Divergent series">diverges</a> for all other values of s. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠ 1. For s = 1, the series is the <a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">harmonic series</a> which diverges to +∞, and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{s\to 1}(s-1)\zeta (s)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo stretchy="false">→</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{s\to 1}(s-1)\zeta (s)=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f618bd444fbcf93ab77acbbb89b07349332c34d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.041ex; height:4.009ex;" alt="{\displaystyle \lim _{s\to 1}(s-1)\zeta (s)=1.}"> Thus the Riemann zeta function is a <a href="/wiki/Meromorphic_function" title="Meromorphic function">meromorphic function</a> on the whole complex plane, which is <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic</a> everywhere except for a <a href="/wiki/Simple_pole" class="mw-redirect" title="Simple pole">simple pole</a> at s = 1 with <a href="/wiki/Residue_(complex_analysis)" title="Residue (complex analysis)">residue</a> 1. <div class="mw-heading mw-heading2"><h2 id="Euler's_product_formula">Euler's product formula</h2>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=2" title="Edit section: Euler's product formula">edit</a>]</div> In 1737, the connection between the zeta function and <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> was discovered by Euler, who <a href="/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function" title="Proof of the Euler product formula for the Riemann zeta function">proved the identity</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> prime</mtext> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81cc405085175957d8728c7ef9c9ce020d6a9ffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.921ex; height:7.176ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}},}"></dd></dl> where, by definition, the left hand side is ζ(s) and the <a href="/wiki/Infinite_product" title="Infinite product">infinite product</a> on the right hand side extends over all prime numbers p (such expressions are called <a href="/wiki/Euler_product" title="Euler product">Euler products</a>): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}={\frac {1}{1-2^{-s}}}\cdot {\frac {1}{1-3^{-s}}}\cdot {\frac {1}{1-5^{-s}}}\cdot {\frac {1}{1-7^{-s}}}\cdot {\frac {1}{1-11^{-s}}}\cdots {\frac {1}{1-p^{-s}}}\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> prime</mtext> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}={\frac {1}{1-2^{-s}}}\cdot {\frac {1}{1-3^{-s}}}\cdot {\frac {1}{1-5^{-s}}}\cdot {\frac {1}{1-7^{-s}}}\cdot {\frac {1}{1-11^{-s}}}\cdots {\frac {1}{1-p^{-s}}}\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d5d41859f464be21d261f9209946d1e5f117b2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.063ex; width:81.555ex; height:6.676ex;" alt="{\displaystyle \prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}={\frac {1}{1-2^{-s}}}\cdot {\frac {1}{1-3^{-s}}}\cdot {\frac {1}{1-5^{-s}}}\cdot {\frac {1}{1-7^{-s}}}\cdot {\frac {1}{1-11^{-s}}}\cdots {\frac {1}{1-p^{-s}}}\cdots }"></dd></dl> Both sides of the Euler product formula converge for Re(s) > 1. The <a href="/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function" title="Proof of the Euler product formula for the Riemann zeta function">proof of Euler's identity</a> uses only the formula for the <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a> and the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>. Since the <a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">harmonic series</a>, obtained when s = 1, diverges, Euler's formula (which becomes Πp <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠p/p − 1⁠) implies that there are <a href="/wiki/Euclid%27s_theorem" title="Euclid's theorem">infinitely many primes</a>.<a href="#cite_note-5">[5]</a> Since the logarithm of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠p/p − 1⁠ is approximately <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠, the formula can also be used to prove the stronger result that the sum of the reciprocals of the primes is infinite. On the other hand, combining that with the <a href="/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">sieve of Eratosthenes</a> shows that the density of the set of primes within the set of positive integers is zero. The Euler product formula can be used to calculate the <a href="/wiki/Asymptotic_density" class="mw-redirect" title="Asymptotic density">asymptotic probability</a> that s randomly selected integers are set-wise <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a>. Intuitively, the probability that any single number is divisible by a prime (or any integer) p is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠. Hence the probability that s numbers are all divisible by this prime is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/ps⁠, and the probability that at least one of them is not is 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/ps⁠. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors n and m if and only if it is divisible by nm, an event which occurs with probability <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/nm⁠). Thus the asymptotic probability that s numbers are coprime is given by a product over all primes, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{p{\text{ prime}}}\left(1-{\frac {1}{p^{s}}}\right)=\left(\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}\right)^{-1}={\frac {1}{\zeta (s)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> prime</mtext> </mrow> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> prime</mtext> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{p{\text{ prime}}}\left(1-{\frac {1}{p^{s}}}\right)=\left(\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}\right)^{-1}={\frac {1}{\zeta (s)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8456bf67df8e20dc977ce04034f6e192504b6078" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.063ex; width:48.294ex; height:8.176ex;" alt="{\displaystyle \prod _{p{\text{ prime}}}\left(1-{\frac {1}{p^{s}}}\right)=\left(\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}\right)^{-1}={\frac {1}{\zeta (s)}}.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Riemann's_functional_equation">Riemann's functional equation</h2>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=3" title="Edit section: Riemann's functional equation">edit</a>]</div> This zeta function satisfies the <a href="/wiki/Functional_equation" title="Functional equation">functional equation</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s)\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>s</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s)\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7538ced199a6bdd1465e39c45bd590b7684d4d78" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.296ex; height:4.843ex;" alt="{\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s)\ ,}"> where Γ(s) is the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>. This is an equality of meromorphic functions valid on the whole <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ(s) has a simple zero at each even negative integer s = −2n, known as the <a href="/wiki/Triviality_(mathematics)" title="Triviality (mathematics)">trivial</a> zeros of ζ(s). When s is an even positive integer, the product sin(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ π s / 2 ⁠) Γ(1 − s) on the right is non-zero because Γ(1 − s) has a simple <a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">pole</a>, which cancels the simple zero of the sine factor. <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style="">Proof of Riemann's functional equation A proof of the functional equation proceeds as follows: We observe that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ s>0\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>s</mi> <mo>></mo> <mn>0</mn> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ s>0\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2391b13f74d677e2e1a239a9dc946a4b572b9e40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.16ex; height:2.509ex;" alt="{\displaystyle \ s>0\ ,}"> then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }x^{{\frac {1}{2}}s-1}e^{-n^{2}\pi x}\ \operatorname {d} x\ =\ {\frac {\ \Gamma \!\left({\frac {s}{2}}\right)\ }{\ n^{s}\ \pi ^{\frac {s}{2}}\ }}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mi>x</mi> </mrow> </msup> <mtext> </mtext> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi mathvariant="normal">Γ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mtext> </mtext> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }x^{{\frac {1}{2}}s-1}e^{-n^{2}\pi x}\ \operatorname {d} x\ =\ {\frac {\ \Gamma \!\left({\frac {s}{2}}\right)\ }{\ n^{s}\ \pi ^{\frac {s}{2}}\ }}~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/599b0c625c312c5d6f9e276b60b48c92dc969bb9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.468ex; height:7.343ex;" alt="{\displaystyle \int _{0}^{\infty }x^{{\frac {1}{2}}s-1}e^{-n^{2}\pi x}\ \operatorname {d} x\ =\ {\frac {\ \Gamma \!\left({\frac {s}{2}}\right)\ }{\ n^{s}\ \pi ^{\frac {s}{2}}\ }}~.}"> As a result, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ s>1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>s</mi> <mo>></mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ s>1\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/058af215387c41d80c387ac3d850aa13b7306afd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.513ex; height:2.176ex;" alt="{\displaystyle \ s>1\ }"> then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\ \Gamma \!\left({\frac {s}{2}}\right)\ \zeta (s)\ }{\ \pi ^{\frac {s}{2}}\ }}\ =\ \sum _{n=1}^{\infty }\ \int _{0}^{\infty }\ x^{{s \over 2}-1}\ e^{-n^{2}\pi x}\ \operatorname {d} x\ =\ \int _{0}^{\infty }x^{{s \over 2}-1}\sum _{n=1}^{\infty }e^{-n^{2}\pi x}\ \operatorname {d} x\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi mathvariant="normal">Γ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mi>x</mi> </mrow> </msup> <mtext> </mtext> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mi>x</mi> </mrow> </msup> <mtext> </mtext> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\ \Gamma \!\left({\frac {s}{2}}\right)\ \zeta (s)\ }{\ \pi ^{\frac {s}{2}}\ }}\ =\ \sum _{n=1}^{\infty }\ \int _{0}^{\infty }\ x^{{s \over 2}-1}\ e^{-n^{2}\pi x}\ \operatorname {d} x\ =\ \int _{0}^{\infty }x^{{s \over 2}-1}\sum _{n=1}^{\infty }e^{-n^{2}\pi x}\ \operatorname {d} x\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bf4caf293a16f2c0e9f41dc319f3ca1e32bf01f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:73.05ex; height:7.343ex;" alt="{\displaystyle {\frac {\ \Gamma \!\left({\frac {s}{2}}\right)\ \zeta (s)\ }{\ \pi ^{\frac {s}{2}}\ }}\ =\ \sum _{n=1}^{\infty }\ \int _{0}^{\infty }\ x^{{s \over 2}-1}\ e^{-n^{2}\pi x}\ \operatorname {d} x\ =\ \int _{0}^{\infty }x^{{s \over 2}-1}\sum _{n=1}^{\infty }e^{-n^{2}\pi x}\ \operatorname {d} x\ ,}"> with the inversion of the limiting processes justified by absolute convergence (hence the stricter requirement on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}">). For convenience, let <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)\ :=\ \sum _{n=1}^{\infty }\ e^{-n^{2}\pi x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>:=</mo> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)\ :=\ \sum _{n=1}^{\infty }\ e^{-n^{2}\pi x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30c9fe2c8867ad04e32baad7423182b6967f2dc9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.563ex; height:6.843ex;" alt="{\displaystyle \psi (x)\ :=\ \sum _{n=1}^{\infty }\ e^{-n^{2}\pi x}}"> which is a special case of the <a href="/wiki/Theta_function" title="Theta function">theta function</a>. Then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)\ =\ {\frac {\pi ^{s \over 2}}{\ \Gamma ({s \over 2})\ }}\ \int _{0}^{\infty }\ x^{{1 \over 2}{s}-1}\ \psi (x)\ \operatorname {d} x~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mrow> <mtext> </mtext> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)\ =\ {\frac {\pi ^{s \over 2}}{\ \Gamma ({s \over 2})\ }}\ \int _{0}^{\infty }\ x^{{1 \over 2}{s}-1}\ \psi (x)\ \operatorname {d} x~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eae9d05989a753a68f37d0efb814ceed313d7d60" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.822ex; height:7.509ex;" alt="{\displaystyle \zeta (s)\ =\ {\frac {\pi ^{s \over 2}}{\ \Gamma ({s \over 2})\ }}\ \int _{0}^{\infty }\ x^{{1 \over 2}{s}-1}\ \psi (x)\ \operatorname {d} x~.}"> By the <a href="/wiki/Poisson_summation_formula" title="Poisson summation formula">Poisson summation formula</a> we have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=-\infty }^{\infty }\ e^{-n^{2}\pi \ x}\ =\ {\frac {1}{\ {\sqrt {x\ }}\ }}\ \sum _{n=-\infty }^{\infty }\ e^{-{\frac {\ n^{2}\pi \ }{x}}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mtext> </mtext> <mi>x</mi> </mrow> </msup> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mtext> </mtext> </mrow> <mi>x</mi> </mfrac> </mrow> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=-\infty }^{\infty }\ e^{-n^{2}\pi \ x}\ =\ {\frac {1}{\ {\sqrt {x\ }}\ }}\ \sum _{n=-\infty }^{\infty }\ e^{-{\frac {\ n^{2}\pi \ }{x}}}\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ca8b40775e90630615fa035d68a5bd8bfb6475" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.44ex; height:6.843ex;" alt="{\displaystyle \sum _{n=-\infty }^{\infty }\ e^{-n^{2}\pi \ x}\ =\ {\frac {1}{\ {\sqrt {x\ }}\ }}\ \sum _{n=-\infty }^{\infty }\ e^{-{\frac {\ n^{2}\pi \ }{x}}}\ ,}"> so that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 2\ \psi (x)+1\ =\ {\frac {1}{\ {\sqrt {x\ }}\ }}\left(\ 2\ \psi \!\left({\frac {1}{x}}\right)+1\ \right)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mi>ψ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>1</mn> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 2\ \psi (x)+1\ =\ {\frac {1}{\ {\sqrt {x\ }}\ }}\left(\ 2\ \psi \!\left({\frac {1}{x}}\right)+1\ \right)~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa97ccc698a283a0a196e2b287b9f29a64cea7a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:40.512ex; height:6.509ex;" alt="{\displaystyle \ 2\ \psi (x)+1\ =\ {\frac {1}{\ {\sqrt {x\ }}\ }}\left(\ 2\ \psi \!\left({\frac {1}{x}}\right)+1\ \right)~.}"> Hence <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{-{\frac {s}{2}}}\ \Gamma \!\left({\frac {s}{2}}\right)\ \zeta (s)\ =\ \int _{0}^{1}\ x^{{\frac {s}{2}}-1}\ \psi (x)\ \operatorname {d} x+\int _{1}^{\infty }x^{{\frac {s}{2}}-1}\psi (x)\ \operatorname {d} x~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mtext> </mtext> <mi mathvariant="normal">Γ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ^{-{\frac {s}{2}}}\ \Gamma \!\left({\frac {s}{2}}\right)\ \zeta (s)\ =\ \int _{0}^{1}\ x^{{\frac {s}{2}}-1}\ \psi (x)\ \operatorname {d} x+\int _{1}^{\infty }x^{{\frac {s}{2}}-1}\psi (x)\ \operatorname {d} x~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa1eeb004fea641971d3cb2a5de7949c046546e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:62.217ex; height:6.176ex;" alt="{\displaystyle \pi ^{-{\frac {s}{2}}}\ \Gamma \!\left({\frac {s}{2}}\right)\ \zeta (s)\ =\ \int _{0}^{1}\ x^{{\frac {s}{2}}-1}\ \psi (x)\ \operatorname {d} x+\int _{1}^{\infty }x^{{\frac {s}{2}}-1}\psi (x)\ \operatorname {d} x~.}"> This is equivalent to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{1}x^{{\frac {s}{2}}-1}\left({\frac {1}{\ {\sqrt {x\ }}\ }}\ \psi \!\left({\frac {1}{x}}\right)+{\frac {1}{\ 2{\sqrt {x\ }}\ }}-{\frac {1}{2}}\ \right)\ \operatorname {d} x+\int _{1}^{\infty }x^{{s \over 2}-1}\psi (x)\ \operatorname {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mi>ψ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}x^{{\frac {s}{2}}-1}\left({\frac {1}{\ {\sqrt {x\ }}\ }}\ \psi \!\left({\frac {1}{x}}\right)+{\frac {1}{\ 2{\sqrt {x\ }}\ }}-{\frac {1}{2}}\ \right)\ \operatorname {d} x+\int _{1}^{\infty }x^{{s \over 2}-1}\psi (x)\ \operatorname {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fc95f25a7b473f8f66dca6c9f901c59b24085d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:67.397ex; height:6.676ex;" alt="{\displaystyle \int _{0}^{1}x^{{\frac {s}{2}}-1}\left({\frac {1}{\ {\sqrt {x\ }}\ }}\ \psi \!\left({\frac {1}{x}}\right)+{\frac {1}{\ 2{\sqrt {x\ }}\ }}-{\frac {1}{2}}\ \right)\ \operatorname {d} x+\int _{1}^{\infty }x^{{s \over 2}-1}\psi (x)\ \operatorname {d} x}"> or <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\ s-1\ }}-{\frac {1}{\ s\ }}+\int _{0}^{1}\ x^{{\frac {s}{2}}-{\frac {3}{2}}}\ \psi \!\left({\frac {1}{\ x\ }}\right)\ \operatorname {d} x+\int _{1}^{\infty }\ x^{{\frac {s}{2}}-1}\ \psi (x)\ \operatorname {d} x~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>s</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mtext> </mtext> <mi>ψ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>x</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\ s-1\ }}-{\frac {1}{\ s\ }}+\int _{0}^{1}\ x^{{\frac {s}{2}}-{\frac {3}{2}}}\ \psi \!\left({\frac {1}{\ x\ }}\right)\ \operatorname {d} x+\int _{1}^{\infty }\ x^{{\frac {s}{2}}-1}\ \psi (x)\ \operatorname {d} x~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d353b74603e90a330efa99d8ee5b9331873c3b70" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:63.15ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{\ s-1\ }}-{\frac {1}{\ s\ }}+\int _{0}^{1}\ x^{{\frac {s}{2}}-{\frac {3}{2}}}\ \psi \!\left({\frac {1}{\ x\ }}\right)\ \operatorname {d} x+\int _{1}^{\infty }\ x^{{\frac {s}{2}}-1}\ \psi (x)\ \operatorname {d} x~.}"> So <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{-{\frac {s}{2}}}\ \Gamma \!\left({\frac {\ s\ }{2}}\right)\ \zeta (s)\ =\ {\frac {1}{\ s(s-1)\ }}+\int _{1}^{\infty }\ \left(x^{-{\frac {s}{2}}-{\frac {1}{2}}}+x^{{\frac {s}{2}}-1}\right)\ \psi (x)\ \operatorname {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mtext> </mtext> <mi mathvariant="normal">Γ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>s</mi> <mtext> </mtext> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ^{-{\frac {s}{2}}}\ \Gamma \!\left({\frac {\ s\ }{2}}\right)\ \zeta (s)\ =\ {\frac {1}{\ s(s-1)\ }}+\int _{1}^{\infty }\ \left(x^{-{\frac {s}{2}}-{\frac {1}{2}}}+x^{{\frac {s}{2}}-1}\right)\ \psi (x)\ \operatorname {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e72bf9d902f6eaa17de289d55056dc6f1c40741e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:68.143ex; height:6.343ex;" alt="{\displaystyle \pi ^{-{\frac {s}{2}}}\ \Gamma \!\left({\frac {\ s\ }{2}}\right)\ \zeta (s)\ =\ {\frac {1}{\ s(s-1)\ }}+\int _{1}^{\infty }\ \left(x^{-{\frac {s}{2}}-{\frac {1}{2}}}+x^{{\frac {s}{2}}-1}\right)\ \psi (x)\ \operatorname {d} x}"> which is convergent for all s, so holds by analytic continuation. Furthermore, note by inspection that the RHS remains the same if s is replaced by 1 − s . Hence <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\ \Gamma \!\left(\ {\frac {s}{2}}\ \right)\ \zeta \!\left(\ s\ \right)\ }{\ \pi ^{{\frac {s}{2}}\ }\ }}\ =\ {\frac {\ \Gamma \!\left(\ {\frac {1}{2}}-{\frac {s}{2}}\ \right)\ \zeta \!\left(\ 1-s\ \right)\ }{\ \pi ^{{\frac {1}{2}}-{\frac {s}{2}}}\ }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi mathvariant="normal">Γ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mi>ζ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mi>s</mi> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mtext> </mtext> </mrow> </msup> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi mathvariant="normal">Γ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mi>ζ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mtext> </mtext> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\ \Gamma \!\left(\ {\frac {s}{2}}\ \right)\ \zeta \!\left(\ s\ \right)\ }{\ \pi ^{{\frac {s}{2}}\ }\ }}\ =\ {\frac {\ \Gamma \!\left(\ {\frac {1}{2}}-{\frac {s}{2}}\ \right)\ \zeta \!\left(\ 1-s\ \right)\ }{\ \pi ^{{\frac {1}{2}}-{\frac {s}{2}}}\ }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/907d8a0f36b4cf1022a4f933004811c00c8a8320" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:42.582ex; height:7.843ex;" alt="{\displaystyle {\frac {\ \Gamma \!\left(\ {\frac {s}{2}}\ \right)\ \zeta \!\left(\ s\ \right)\ }{\ \pi ^{{\frac {s}{2}}\ }\ }}\ =\ {\frac {\ \Gamma \!\left(\ {\frac {1}{2}}-{\frac {s}{2}}\ \right)\ \zeta \!\left(\ 1-s\ \right)\ }{\ \pi ^{{\frac {1}{2}}-{\frac {s}{2}}}\ }}}"> which is the functional equation attributed to <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a>.<a href="#cite_note-6">[6]</a> The functional equation above can now be obtained using the <a href="/wiki/Multiplication_theorem#Gamma_function–Legendre_formula" title="Multiplication theorem">duplication formula</a> for the gamma function. </div> The functional equation was established by Riemann in his 1859 paper "<a href="/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude" title="On the Number of Primes Less Than a Given Magnitude">On the Number of Primes Less Than a Given Magnitude</a>" and used to construct the analytic continuation in the first place. <div class="mw-heading mw-heading3"><h3 id="Equivalencies">Equivalencies</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=4" title="Edit section: Equivalencies">edit</a>]</div> An equivalent relationship had been conjectured by Euler over a hundred years earlier, in 1749, for the <a href="/wiki/Dirichlet_eta_function" title="Dirichlet eta function">Dirichlet eta function</a> (the alternating zeta function): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta (s)\ =\ \sum _{n=1}^{\infty }{\frac {\;(-1)^{n+1}}{\ n^{s}}}=\left(1-{2^{1-s}}\right)\ \zeta (s)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mtext> </mtext> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta (s)\ =\ \sum _{n=1}^{\infty }{\frac {\;(-1)^{n+1}}{\ n^{s}}}=\left(1-{2^{1-s}}\right)\ \zeta (s)~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff1608984b5a76e49f2c9b3f7cb9270d33c6e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.726ex; height:7.009ex;" alt="{\displaystyle \eta (s)\ =\ \sum _{n=1}^{\infty }{\frac {\;(-1)^{n+1}}{\ n^{s}}}=\left(1-{2^{1-s}}\right)\ \zeta (s)~.}"> Incidentally, this relation gives an equation for calculating ζ(s) in the region 0 < ℛℯ(s) < 1 , i.e. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)={\frac {1}{\;1-2^{1-s}\ }}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{\;n^{s}\ }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thickmathspace" /> <mn>1</mn> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mtext> </mtext> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mspace width="thickmathspace" /> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mtext> </mtext> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)={\frac {1}{\;1-2^{1-s}\ }}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{\;n^{s}\ }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1157febfe010c57cc0c7ffd04285410819c04bd3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.489ex; height:7.009ex;" alt="{\displaystyle \zeta (s)={\frac {1}{\;1-2^{1-s}\ }}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{\;n^{s}\ }}}"> where the η-series is <a href="/wiki/Convergent_series" title="Convergent series">convergent</a> (albeit <a href="/wiki/Absolute_convergence" title="Absolute convergence">non-absolutely</a>) in the larger half-plane s > 0 (for a more detailed survey on the history of the functional equation, see e.g. Blagouchine<a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a>). Riemann also found a <a href="/wiki/Symmetry" title="Symmetry">symmetric</a> version of the functional equation applying to the ξ-function: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi (s)\ =\ {\frac {1}{2}}\pi ^{-{\frac {s}{2}}}\ s(s-1)\ \Gamma \!\left({\frac {s}{2}}\right)\ \zeta (s)\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mtext> </mtext> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> <mi mathvariant="normal">Γ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi (s)\ =\ {\frac {1}{2}}\pi ^{-{\frac {s}{2}}}\ s(s-1)\ \Gamma \!\left({\frac {s}{2}}\right)\ \zeta (s)\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6e84309e83cec06e8c5364b09e027a251cd3ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.108ex; height:5.176ex;" alt="{\displaystyle \xi (s)\ =\ {\frac {1}{2}}\pi ^{-{\frac {s}{2}}}\ s(s-1)\ \Gamma \!\left({\frac {s}{2}}\right)\ \zeta (s)\ ,}"> which satisfies: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi (s)=\xi (1-s)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ξ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi (s)=\xi (1-s)~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0333067155def9c5fda4163779866eeac9613707" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.188ex; height:2.843ex;" alt="{\displaystyle \xi (s)=\xi (1-s)~.}"> (Riemann's <a href="/wiki/Riemann_%CE%9E_function" class="mw-redirect" title="Riemann Ξ function">original ξ(t)</a> was slightly different.) The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \pi ^{-s/2}\ \Gamma (s/2)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \pi ^{-s/2}\ \Gamma (s/2)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e496f302d4be9d53af2c66fcb48afcce0b7ede96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.679ex; height:3.343ex;" alt="{\displaystyle \ \pi ^{-s/2}\ \Gamma (s/2)\ }"> factor was not well-understood at the time of Riemann, until <a href="/wiki/John_Tate_(mathematician)" title="John Tate (mathematician)">John Tate</a>'s (1950) <a href="/wiki/Tate%27s_thesis" title="Tate's thesis">thesis</a>, in which it was shown that this so-called "Gamma factor" is in fact the <a href="/w/index.php?title=Local_L-factor&action=edit&redlink=1" class="new" title="Local L-factor (page does not exist)">local L-factor</a> corresponding to the <a href="/wiki/Archimedean_place" class="mw-redirect" title="Archimedean place">Archimedean place</a>, the other factors in the Euler product expansion being the local L-factors of the non-Archimedean places. <div class="mw-heading mw-heading2"><h2 id="Zeros,_the_critical_line,_and_the_Riemann_hypothesis">Zeros, the critical line, and the Riemann hypothesis</h2>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=5" title="Edit section: Zeros, the critical line, and the Riemann hypothesis">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Riemann0xf4240.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Riemann0xf4240.png/720px-Riemann0xf4240.png" decoding="async" width="720" height="405" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Riemann0xf4240.png/1080px-Riemann0xf4240.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Riemann0xf4240.png/1440px-Riemann0xf4240.png 2x" data-file-width="7680" data-file-height="4320" /></a><figcaption>Riemann zeta spiral along the critical line from height 999000 to a million (from red to violet)</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Zero-free_region_for_the_Riemann_zeta-function.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Zero-free_region_for_the_Riemann_zeta-function.svg/300px-Zero-free_region_for_the_Riemann_zeta-function.svg.png" decoding="async" width="300" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Zero-free_region_for_the_Riemann_zeta-function.svg/450px-Zero-free_region_for_the_Riemann_zeta-function.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Zero-free_region_for_the_Riemann_zeta-function.svg/600px-Zero-free_region_for_the_Riemann_zeta-function.svg.png 2x" data-file-width="619" data-file-height="569" /></a><figcaption>The Riemann zeta function has no zeros to the right of σ = 1 or (apart from the trivial zeros) to the left of σ = 0 (nor can the zeros lie too close to those lines). Furthermore, the non-trivial zeros are symmetric about the real axis and the line σ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ and, according to the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>, they all lie on the line σ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Zeta_polar.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Zeta_polar.svg/300px-Zeta_polar.svg.png" decoding="async" width="300" height="257" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Zeta_polar.svg/450px-Zeta_polar.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Zeta_polar.svg/600px-Zeta_polar.svg.png 2x" data-file-width="560" data-file-height="480" /></a><figcaption>This image shows a plot of the Riemann zeta function along the critical line for real values of t running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:RiemannCriticalLine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/RiemannCriticalLine.svg/300px-RiemannCriticalLine.svg.png" decoding="async" width="300" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/RiemannCriticalLine.svg/450px-RiemannCriticalLine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/RiemannCriticalLine.svg/600px-RiemannCriticalLine.svg.png 2x" data-file-width="933" data-file-height="434" /></a><figcaption>The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Zeta1000_1005.webm/220px--Zeta1000_1005.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="220" data-durationhint="51" data-mwtitle="Zeta1000_1005.webm" data-mwprovider="wikimediacommons" resource="/wiki/File:Zeta1000_1005.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9a/Zeta1000_1005.webm/Zeta1000_1005.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="480" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9a/Zeta1000_1005.webm/Zeta1000_1005.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="720" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/9/9a/Zeta1000_1005.webm" type="video/webm; codecs="vp8"" data-width="2000" data-height="2000" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9a/Zeta1000_1005.webm/Zeta1000_1005.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1080" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9a/Zeta1000_1005.webm/Zeta1000_1005.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="240" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9a/Zeta1000_1005.webm/Zeta1000_1005.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="360" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9a/Zeta1000_1005.webm/Zeta1000_1005.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="360" data-height="360" /></video><figcaption>Animation showing the Riemann zeta function along the critical line. Zeta(1/2 + I y) for y ranging from 1000 to 1005.</figcaption></figure> The functional equation shows that the Riemann zeta function has zeros at −2, −4,.... These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠πs/2⁠ being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields important results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{s\in \mathbb {C} :0<\operatorname {Re} (s)<1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>s</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>:</mo> <mn>0</mn> <mo><</mo> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{s\in \mathbb {C} :0<\operatorname {Re} (s)<1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bd2e09ccf28288b6a9bc02acff20047afeb7454" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.036ex; height:2.843ex;" alt="{\displaystyle \{s\in \mathbb {C} :0<\operatorname {Re} (s)<1\}}">, which is called the critical strip. The set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{s\in \mathbb {C} :\operatorname {Re} (s)=1/2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>s</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>:</mo> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{s\in \mathbb {C} :\operatorname {Re} (s)=1/2\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f83774c1d820095a11751431fe11345673ef87b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.1ex; height:2.843ex;" alt="{\displaystyle \{s\in \mathbb {C} :\operatorname {Re} (s)=1/2\}}"> is called the critical line. The <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>, considered one of the greatest unsolved problems in mathematics, asserts that all non-trivial zeros are on the critical line. In 1989, Conrey proved that more than 40% of the non-trivial zeros of the Riemann zeta function are on the critical line.<a href="#cite_note-9">[9]</a> For the Riemann zeta function on the critical line, see <a href="/wiki/Z_function" title="Z function">Z-function</a>. <table class="wikitable"> <caption>First few nontrivial zeros<a href="#cite_note-10">[10]</a><a href="#cite_note-11">[11]</a> </caption> <tbody><tr> <th>Zero </th></tr> <tr> <td>1/2 ± 14.134725 i </td></tr> <tr> <td>1/2 ± 21.022040 i </td></tr> <tr> <td>1/2 ± 25.010858 i </td></tr> <tr> <td>1/2 ± 30.424876 i </td></tr> <tr> <td>1/2 ± 32.935062 i </td></tr> <tr> <td>1/2 ± 37.586178 i </td></tr> <tr> <td>1/2 ± 40.918719 i </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Number_of_zeros_in_the_critical_strip">Number of zeros in the critical strip</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=6" title="Edit section: Number of zeros in the critical strip">edit</a>]</div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(T)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891f08bfa8e782f2db0a06b00aa11668b1614537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.509ex; height:2.843ex;" alt="{\displaystyle N(T)}"> be the number of zeros of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd45922057e4d7a5718ce5ed703ab493c63897a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.995ex; height:2.843ex;" alt="{\displaystyle \zeta (s)}"> in the critical strip <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\operatorname {Re} (s)<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\operatorname {Re} (s)<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32b9718e606940f63859292d9607b611a545b9d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.165ex; height:2.843ex;" alt="{\displaystyle 0<\operatorname {Re} (s)<1}">, whose imaginary parts are in the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\operatorname {Im} (s)<T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\operatorname {Im} (s)<T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f120c4a75cf6bb318f96e345d62005a60dfa8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.671ex; height:2.843ex;" alt="{\displaystyle 0<\operatorname {Im} (s)<T}">. Trudgian proved that, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T>e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>></mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T>e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4177c581663e58522b046e394e064040d09c6071" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.818ex; height:2.176ex;" alt="{\displaystyle T>e}">, then<a href="#cite_note-12">[12]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|N(T)-{\frac {T}{2\pi }}\log {\frac {T}{2\pi e}}\right|\leq 0.112\log T+0.278\log \log T+3.385+{\frac {0.2}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mn>0.112</mn> <mi>log</mi> <mo>⁡</mo> <mi>T</mi> <mo>+</mo> <mn>0.278</mn> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>T</mi> <mo>+</mo> <mn>3.385</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>0.2</mn> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|N(T)-{\frac {T}{2\pi }}\log {\frac {T}{2\pi e}}\right|\leq 0.112\log T+0.278\log \log T+3.385+{\frac {0.2}{T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/375e5d6b6af31718215474d76d5dc700980a9973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:66.575ex; height:5.509ex;" alt="{\displaystyle \left|N(T)-{\frac {T}{2\pi }}\log {\frac {T}{2\pi e}}\right|\leq 0.112\log T+0.278\log \log T+3.385+{\frac {0.2}{T}}}">.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="The_Hardy–Littlewood_conjectures">The Hardy–Littlewood conjectures</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=7" title="Edit section: The Hardy–Littlewood conjectures">edit</a>]</div> In 1914, <a href="/wiki/G._H._Hardy" title="G. H. Hardy">G. H. Hardy</a> proved that ζ (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ + it) has infinitely many real zeros.<a href="#cite_note-13">[13]</a><a href="#cite_note-14">[14]</a> Hardy and <a href="/wiki/John_Edensor_Littlewood" title="John Edensor Littlewood">J. E. Littlewood</a> formulated two conjectures on the density and distance between the zeros of ζ (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ + it) on intervals of large positive real numbers. In the following, N(T) is the total number of real zeros and N0(T) the total number of zeros of odd order of the function ζ (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ + it) lying in the interval (0, T]. <div><ol><li>For any ε > 0, there exists a T0(ε) > 0 such that when <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\geq T_{0}(\varepsilon )\quad {\text{ and }}\quad H=T^{{\frac {1}{4}}+\varepsilon },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>≥</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="1em" /> <mi>H</mi> <mo>=</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\geq T_{0}(\varepsilon )\quad {\text{ and }}\quad H=T^{{\frac {1}{4}}+\varepsilon },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b72a56f6b0bc1ad48948ae4087520a5649224474" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.903ex; height:4.009ex;" alt="{\displaystyle T\geq T_{0}(\varepsilon )\quad {\text{ and }}\quad H=T^{{\frac {1}{4}}+\varepsilon },}"></dd></dl> the interval (T, T + H] contains a zero of odd order.</li><li>For any ε > 0, there exists a T0(ε) > 0 and cε > 0 such that the inequality <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{0}(T+H)-N_{0}(T)\geq c_{\varepsilon }H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>+</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{0}(T+H)-N_{0}(T)\geq c_{\varepsilon }H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb3f056dd54fb9048ccdb162de032e8f5b86ca4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.644ex; height:2.843ex;" alt="{\displaystyle N_{0}(T+H)-N_{0}(T)\geq c_{\varepsilon }H}"></dd></dl> holds when <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\geq T_{0}(\varepsilon )\quad {\text{ and }}\quad H=T^{{\frac {1}{2}}+\varepsilon }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>≥</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="1em" /> <mi>H</mi> <mo>=</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\geq T_{0}(\varepsilon )\quad {\text{ and }}\quad H=T^{{\frac {1}{2}}+\varepsilon }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e546a591646c3a6e0a8bb464f6852877c55aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.903ex; height:4.009ex;" alt="{\displaystyle T\geq T_{0}(\varepsilon )\quad {\text{ and }}\quad H=T^{{\frac {1}{2}}+\varepsilon }.}"></dd></dl></li></ol></div> These two conjectures opened up new directions in the investigation of the Riemann zeta function. <div class="mw-heading mw-heading3"><h3 id="Zero-free_region">Zero-free region</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=8" title="Edit section: Zero-free region">edit</a>]</div> The location of the Riemann zeta function's zeros is of great importance in number theory. The <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a> is equivalent to the fact that there are no zeros of the zeta function on the Re(s) = 1 line.<a href="#cite_note-Diamond1982-15">[15]</a> A better result<a href="#cite_note-16">[16]</a> that follows from an effective form of <a href="/wiki/Vinogradov%27s_mean-value_theorem" title="Vinogradov's mean-value theorem">Vinogradov's mean-value theorem</a> is that ζ (σ + it) ≠ 0 whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \geq 1-{\frac {1}{57.54(\log {|t|})^{\frac {2}{3}}(\log {\log {|t|}})^{\frac {1}{3}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>≥</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>57.54</mn> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </msup> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma \geq 1-{\frac {1}{57.54(\log {|t|})^{\frac {2}{3}}(\log {\log {|t|}})^{\frac {1}{3}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/070e5d70379dd28aafbf8671116b61070abda82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:35.997ex; height:7.176ex;" alt="{\displaystyle \sigma \geq 1-{\frac {1}{57.54(\log {|t|})^{\frac {2}{3}}(\log {\log {|t|}})^{\frac {1}{3}}}}}"> and |t| ≥ 3. In 2015, Mossinghoff and Trudgian proved<a href="#cite_note-17">[17]</a> that zeta has no zeros in the region <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \geq 1-{\frac {1}{5.573412\log |t|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>≥</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>5.573412</mn> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma \geq 1-{\frac {1}{5.573412\log |t|}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2858ec99d012142fe009fa2da224426062f9fb62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.93ex; height:6.009ex;" alt="{\displaystyle \sigma \geq 1-{\frac {1}{5.573412\log |t|}}}"></dd></dl> for |t| ≥ 2. This is the largest known zero-free region in the critical strip for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3.06\cdot 10^{10}<|t|<\exp(10151.5)\approx 5.5\cdot 10^{4408}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3.06</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>10151.5</mn> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>5.5</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4408</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3.06\cdot 10^{10}<|t|<\exp(10151.5)\approx 5.5\cdot 10^{4408}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52a848aef22bbf86f863eb6aa0c32f358103589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.922ex; height:3.176ex;" alt="{\displaystyle 3.06\cdot 10^{10}<|t|<\exp(10151.5)\approx 5.5\cdot 10^{4408}}">. The strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound <a href="/wiki/Riemann_hypothesis#Consequences" title="Riemann hypothesis">consequences</a> in the theory of numbers. <div class="mw-heading mw-heading3"><h3 id="Other_results">Other results</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=9" title="Edit section: Other results">edit</a>]</div> It is known that there are infinitely many zeros on the critical line. <a href="/wiki/John_Edensor_Littlewood" title="John Edensor Littlewood">Littlewood</a> showed that if the sequence (γn) contains the imaginary parts of all zeros in the <a href="/wiki/Upper_half-plane" title="Upper half-plane">upper half-plane</a> in ascending order, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\rightarrow \infty }\left(\gamma _{n+1}-\gamma _{n}\right)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty }\left(\gamma _{n+1}-\gamma _{n}\right)=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e3005b4211360d5f3053b8885baf0a7449baee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.163ex; height:3.843ex;" alt="{\displaystyle \lim _{n\rightarrow \infty }\left(\gamma _{n+1}-\gamma _{n}\right)=0.}"></dd></dl> The <a href="/wiki/Critical_line_theorem" class="mw-redirect" title="Critical line theorem">critical line theorem</a> asserts that a positive proportion of the nontrivial zeros lies on the critical line. (The Riemann hypothesis would imply that this proportion is 1.) In the critical strip, the zero with smallest non-negative imaginary part is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ + 14.13472514...i (<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A058303" class="extiw" title="oeis:A058303">A058303</a>). The fact that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)={\overline {\zeta ({\overline {s}})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)={\overline {\zeta ({\overline {s}})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b09034e8f5f6df4406385816e4a6f7109e19cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.318ex; height:3.676ex;" alt="{\displaystyle \zeta (s)={\overline {\zeta ({\overline {s}})}}}"></dd></dl> for all complex s ≠ 1 implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line Re(s) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠. It is also known that no zeros lie on the line with real part 1. <div class="mw-heading mw-heading2"><h2 id="Specific_values">Specific values</h2>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=10" title="Edit section: Specific values">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Particular_values_of_the_Riemann_zeta_function" title="Particular values of the Riemann zeta function">Particular values of the Riemann zeta function</a></div> For any positive even integer 2n, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (2n)={\frac {|{B_{2n}}|(2\pi )^{2n}}{2(2n)!}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (2n)={\frac {|{B_{2n}}|(2\pi )^{2n}}{2(2n)!}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7603e9f4c7ce96528feb82f7a3b2b50358a91256" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.485ex; height:6.676ex;" alt="{\displaystyle \zeta (2n)={\frac {|{B_{2n}}|(2\pi )^{2n}}{2(2n)!}},}"> where B2n is the 2n-th <a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli number</a>. For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraic K-theory of the integers; see <a href="/wiki/Special_values_of_L-functions" title="Special values of L-functions">Special values of L-functions</a>. For nonpositive integers, one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (-n)=-{\frac {B_{n+1}}{n+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (-n)=-{\frac {B_{n+1}}{n+1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/745d83a9305e85e80ee607b2d1778f08488584ae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.248ex; height:5.509ex;" alt="{\displaystyle \zeta (-n)=-{\frac {B_{n+1}}{n+1}}}"> for n ≥ 0 (using the convention that B1 = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠). In particular, ζ vanishes at the negative even integers because Bm = 0 for all odd m other than 1. These are the so-called "trivial zeros" of the zeta function. Via <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a>, one can show that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (-1)=-{\tfrac {1}{12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (-1)=-{\tfrac {1}{12}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/242b35184f3c25d8af6a1fa0e992a015ca8b07f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.262ex; height:3.509ex;" alt="{\displaystyle \zeta (-1)=-{\tfrac {1}{12}}}"> This gives a pretext for assigning a finite value to the divergent series <a href="/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯">1 + 2 + 3 + 4 + ⋯</a>, which has been used in certain contexts (<a href="/wiki/Ramanujan_summation" title="Ramanujan summation">Ramanujan summation</a>) such as <a href="/wiki/String_theory" title="String theory">string theory</a>.<a href="#cite_note-polchinski-18">[18]</a> Analogously, the particular value <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (0)=-{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (0)=-{\tfrac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f12d5ccd326048205102bbe612a2d639d52c7f3b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.632ex; height:3.509ex;" alt="{\displaystyle \zeta (0)=-{\tfrac {1}{2}}}"> can be viewed as assigning a finite result to the divergent series <a href="/wiki/1_%2B_1_%2B_1_%2B_1_%2B_%E2%8B%AF" title="1 + 1 + 1 + 1 + ⋯">1 + 1 + 1 + 1 + ⋯</a>. The value <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta {\bigl (}{\tfrac {1}{2}}{\bigr )}=-1.46035450880958681288\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mn>1.46035450880958681288</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta {\bigl (}{\tfrac {1}{2}}{\bigr )}=-1.46035450880958681288\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/530578bc4b4fe11508d90842bf6fc369e3a8eb5f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:37.958ex; height:3.509ex;" alt="{\displaystyle \zeta {\bigl (}{\tfrac {1}{2}}{\bigr )}=-1.46035450880958681288\ldots }"> is employed in calculating kinetic boundary layer problems of linear kinetic equations.<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a> Although <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (1)=1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (1)=1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e80a209099a8bac2fe5d5f80e5183b9cf5c0ee4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.888ex; height:3.676ex;" alt="{\displaystyle \zeta (1)=1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\cdots }"> diverges, its <a href="/wiki/Cauchy_principal_value" title="Cauchy principal value">Cauchy principal value</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{\varepsilon \to 0}{\frac {\zeta (1+\varepsilon )+\zeta (1-\varepsilon )}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\varepsilon \to 0}{\frac {\zeta (1+\varepsilon )+\zeta (1-\varepsilon )}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe362a14970085cef83fab635bd8f487f92a7c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.276ex; height:5.843ex;" alt="{\displaystyle \lim _{\varepsilon \to 0}{\frac {\zeta (1+\varepsilon )+\zeta (1-\varepsilon )}{2}}}"> exists and is equal to the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a> γ = 0.5772....<a href="#cite_note-Sondow1998-21">[21]</a> The demonstration of the particular value <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (2)=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (2)=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c47997aa5335da52ab90564eed52075417910256" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.001ex; height:6.176ex;" alt="{\displaystyle \zeta (2)=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}}"> is known as the <a href="/wiki/Basel_problem" title="Basel problem">Basel problem</a>. The reciprocal of this sum answers the question: What is the probability that two numbers selected at random are <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">relatively prime</a>?<a href="#cite_note-22">[22]</a> The value <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (3)=1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots =1.202056903159594285399...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>1.202056903159594285399...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (3)=1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots =1.202056903159594285399...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b34718e242d38fc199c25e009a393cf3184d1de2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:56.938ex; height:5.676ex;" alt="{\displaystyle \zeta (3)=1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots =1.202056903159594285399...}"> is <a href="/wiki/Ap%C3%A9ry%27s_constant" title="Apéry's constant">Apéry's constant</a>. Taking the limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\rightarrow +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\rightarrow +\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a97aa3ada267c5151057ba9d4ce99bb2872c566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.836ex; height:2.176ex;" alt="{\displaystyle s\rightarrow +\infty }"> through the real numbers, one obtains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (+\infty )=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (+\infty )=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71422328c935dd7551a76a947bcc3526bb569fdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.297ex; height:2.843ex;" alt="{\displaystyle \zeta (+\infty )=1}">. But at <a href="/wiki/Complex_infinity" class="mw-redirect" title="Complex infinity">complex infinity</a> on the <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a> the zeta function has an <a href="/wiki/Essential_singularity" title="Essential singularity">essential singularity</a>.<a href="#cite_note-:0-2">[2]</a> <div class="mw-heading mw-heading2"><h2 id="Various_properties">Various properties</h2>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=11" title="Edit section: Various properties">edit</a>]</div> For sums involving the zeta function at integer and half-integer values, see <a href="/wiki/Rational_zeta_series" title="Rational zeta series">rational zeta series</a>. <div class="mw-heading mw-heading3"><h3 id="Reciprocal">Reciprocal</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=12" title="Edit section: Reciprocal">edit</a>]</div> The reciprocal of the zeta function may be expressed as a <a href="/wiki/Dirichlet_series" title="Dirichlet series">Dirichlet series</a> over the <a href="/wiki/M%C3%B6bius_function" title="Möbius function">Möbius function</a> μ(n): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44321887183e072812ddd22b8ed103e08ced1e0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.113ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}}"></dd></dl> for every complex number s with real part greater than 1. There are a number of similar relations involving various well-known <a href="/wiki/Multiplicative_function" title="Multiplicative function">multiplicative functions</a>; these are given in the article on the <a href="/wiki/Dirichlet_series" title="Dirichlet series">Dirichlet series</a>. The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠. <div class="mw-heading mw-heading3"><h3 id="Universality">Universality</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=13" title="Edit section: Universality">edit</a>]</div> The critical strip of the Riemann zeta function has the remarkable property of universality. This <a href="/wiki/Zeta_function_universality" title="Zeta function universality">zeta function universality</a> states that there exists some location on the critical strip that approximates any <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic function</a> arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable. The first proof of universality was provided by Sergei Mikhailovitch Voronin in 1975.<a href="#cite_note-23">[23]</a> More recent work has included <a href="/wiki/Zeta_function_universality#Effective_universality" title="Zeta function universality">effective</a> versions of Voronin's theorem<a href="#cite_note-24">[24]</a> and <a href="/wiki/Zeta_function_universality#Universality_of_other_zeta_functions" title="Zeta function universality">extending</a> it to <a href="/wiki/Dirichlet_L-function" title="Dirichlet L-function">Dirichlet L-functions</a>.<a href="#cite_note-25">[25]</a><a href="#cite_note-26">[26]</a> <div class="mw-heading mw-heading3"><h3 id="Estimates_of_the_maximum_of_the_modulus_of_the_zeta_function">Estimates of the maximum of the modulus of the zeta function</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=14" title="Edit section: Estimates of the maximum of the modulus of the zeta function">edit</a>]</div> Let the functions F(T;H) and G(s0;Δ) be defined by the equalities <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(T;H)=\max _{|t-T|\leq H}\left|\zeta \left({\tfrac {1}{2}}+it\right)\right|,\qquad G(s_{0};\Delta )=\max _{|s-s_{0}|\leq \Delta }|\zeta (s)|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>;</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mo>−</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>H</mi> </mrow> </munder> <mrow> <mo>|</mo> <mrow> <mi>ζ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mi>i</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>;</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi mathvariant="normal">Δ</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(T;H)=\max _{|t-T|\leq H}\left|\zeta \left({\tfrac {1}{2}}+it\right)\right|,\qquad G(s_{0};\Delta )=\max _{|s-s_{0}|\leq \Delta }|\zeta (s)|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a20a242960523633d776be2041a97279d8dbe02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.95ex; height:4.843ex;" alt="{\displaystyle F(T;H)=\max _{|t-T|\leq H}\left|\zeta \left({\tfrac {1}{2}}+it\right)\right|,\qquad G(s_{0};\Delta )=\max _{|s-s_{0}|\leq \Delta }|\zeta (s)|.}"></dd></dl> Here T is a sufficiently large positive number, 0 < H ≪ log log T, s0 = σ0 + iT, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ ≤ σ0 ≤ 1, 0 < Δ < <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠. Estimating the values F and G from below shows, how large (in modulus) values ζ(s) can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip 0 ≤ Re(s) ≤ 1. The case H ≫ log log T was studied by <a href="/wiki/Kanakanahalli_Ramachandra" title="Kanakanahalli Ramachandra">Kanakanahalli Ramachandra</a>; the case Δ > c, where c is a sufficiently large constant, is trivial. <a href="/wiki/Anatolii_Alexeevitch_Karatsuba" class="mw-redirect" title="Anatolii Alexeevitch Karatsuba">Anatolii Karatsuba</a> proved,<a href="#cite_note-27">[27]</a><a href="#cite_note-28">[28]</a> in particular, that if the values H and Δ exceed certain sufficiently small constants, then the estimates <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(T;H)\geq T^{-c_{1}},\qquad G(s_{0};\Delta )\geq T^{-c_{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>;</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>,</mo> <mspace width="2em" /> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>;</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(T;H)\geq T^{-c_{1}},\qquad G(s_{0};\Delta )\geq T^{-c_{2}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c01506233e6725116b2bb19cfb60c99635deb5e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.106ex; height:3.009ex;" alt="{\displaystyle F(T;H)\geq T^{-c_{1}},\qquad G(s_{0};\Delta )\geq T^{-c_{2}},}"></dd></dl> hold, where c1 and c2 are certain absolute constants. <div class="mw-heading mw-heading3"><h3 id="The_argument_of_the_Riemann_zeta_function">The argument of the Riemann zeta function</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=15" title="Edit section: The argument of the Riemann zeta function">edit</a>]</div> The function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(t)={\frac {1}{\pi }}\arg {\zeta \left({\tfrac {1}{2}}+it\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mi>arg</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mi>i</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(t)={\frac {1}{\pi }}\arg {\zeta \left({\tfrac {1}{2}}+it\right)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/114f40b882f5f7cb1c5709ac5ce5bcbf848dc4f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.178ex; height:5.176ex;" alt="{\displaystyle S(t)={\frac {1}{\pi }}\arg {\zeta \left({\tfrac {1}{2}}+it\right)}}"></dd></dl> is called the <a href="/wiki/Complex_argument" class="mw-redirect" title="Complex argument">argument</a> of the Riemann zeta function. Here arg ζ(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ + it) is the increment of an arbitrary continuous branch of arg ζ(s) along the broken line joining the points 2, 2 + it and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ + it. There are some theorems on properties of the function S(t). Among those results<a href="#cite_note-29">[29]</a><a href="#cite_note-30">[30]</a> are the <a href="/wiki/Mean_value_theorems_for_definite_integrals" class="mw-redirect" title="Mean value theorems for definite integrals">mean value theorems</a> for S(t) and its first integral <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}(t)=\int _{0}^{t}S(u)\,\mathrm {d} u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>S</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}(t)=\int _{0}^{t}S(u)\,\mathrm {d} u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8680a253ffbb7a376bdd76b07c696bd984c0201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.551ex; height:6.176ex;" alt="{\displaystyle S_{1}(t)=\int _{0}^{t}S(u)\,\mathrm {d} u}"></dd></dl> on intervals of the real line, and also the theorem claiming that every interval (T, T + H] for <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\geq T^{{\frac {27}{82}}+\varepsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>≥</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>27</mn> <mn>82</mn> </mfrac> </mrow> <mo>+</mo> <mi>ε</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\geq T^{{\frac {27}{82}}+\varepsilon }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deb900d659d59ff997c06e31da1601b3a66d1ccf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.33ex; height:3.843ex;" alt="{\displaystyle H\geq T^{{\frac {27}{82}}+\varepsilon }}"></dd></dl> contains at least <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H{\sqrt[{3}]{\ln T}}e^{-c{\sqrt {\ln \ln T}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ln</mi> <mo>⁡</mo> <mi>ln</mi> <mo>⁡</mo> <mi>T</mi> </msqrt> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H{\sqrt[{3}]{\ln T}}e^{-c{\sqrt {\ln \ln T}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2a08b909b3fbdc065cdc21c25e4469fb8136817" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.311ex; height:3.343ex;" alt="{\displaystyle H{\sqrt[{3}]{\ln T}}e^{-c{\sqrt {\ln \ln T}}}}"></dd></dl> points where the function S(t) changes sign. Earlier similar results were obtained by <a href="/wiki/Atle_Selberg" title="Atle Selberg">Atle Selberg</a> for the case <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\geq T^{{\frac {1}{2}}+\varepsilon }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>≥</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\geq T^{{\frac {1}{2}}+\varepsilon }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e53e06ac6d1832b6c40070c4582ec495a38a6678" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.309ex; height:3.676ex;" alt="{\displaystyle H\geq T^{{\frac {1}{2}}+\varepsilon }.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Representations">Representations</h2>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=16" title="Edit section: Representations">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Dirichlet_series">Dirichlet series</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=17" title="Edit section: Dirichlet series">edit</a>]</div> An extension of the area of convergence can be obtained by rearranging the original series.<a href="#cite_note-Knopp-31">[31]</a> The series <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=1}^{\infty }\left({\frac {n}{(n+1)^{s}}}-{\frac {n-s}{n^{s}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mi>s</mi> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=1}^{\infty }\left({\frac {n}{(n+1)^{s}}}-{\frac {n-s}{n^{s}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d536ef583045ec0efbb1d4b5bdf478125e79440c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.622ex; height:6.843ex;" alt="{\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=1}^{\infty }\left({\frac {n}{(n+1)^{s}}}-{\frac {n-s}{n^{s}}}\right)}"></dd></dl> converges for Re(s) > 0, while <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=1}^{\infty }{\frac {n(n+1)}{2}}\left({\frac {2n+3+s}{(n+1)^{s+2}}}-{\frac {2n-1-s}{n^{s+2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mi>s</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>s</mi> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=1}^{\infty }{\frac {n(n+1)}{2}}\left({\frac {2n+3+s}{(n+1)^{s+2}}}-{\frac {2n-1-s}{n^{s+2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2f44909dd2c6e7378b7bcbd6baa45fcd978fa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:55.892ex; height:6.843ex;" alt="{\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=1}^{\infty }{\frac {n(n+1)}{2}}\left({\frac {2n+3+s}{(n+1)^{s+2}}}-{\frac {2n-1-s}{n^{s+2}}}\right)}"></dd></dl> converge even for Re(s) > −1. In this way, the area of convergence can be extended to Re(s) > −k for any negative integer −k. The recurrence connection is clearly visible from the expression valid for Re(s) > −2 enabling further expansion by integration by parts. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\zeta (s)=&1+{\frac {1}{s-1}}-{\frac {s}{2!}}[\zeta (s+1)-1]\\-&{\frac {s(s+1)}{3!}}[\zeta (s+2)-1]\\&-{\frac {s(s+1)(s+2)}{3!}}\sum _{n=1}^{\infty }\int _{0}^{1}{\frac {t^{3}dt}{(n+t)^{s+3}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> <mtd> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>+</mo> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\zeta (s)=&1+{\frac {1}{s-1}}-{\frac {s}{2!}}[\zeta (s+1)-1]\\-&{\frac {s(s+1)}{3!}}[\zeta (s+2)-1]\\&-{\frac {s(s+1)(s+2)}{3!}}\sum _{n=1}^{\infty }\int _{0}^{1}{\frac {t^{3}dt}{(n+t)^{s+3}}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86c83010f166d68a4a9811dfd6e4e8cb5f1d746d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.426ex; margin-bottom: -0.245ex; width:44.63ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}\zeta (s)=&1+{\frac {1}{s-1}}-{\frac {s}{2!}}[\zeta (s+1)-1]\\-&{\frac {s(s+1)}{3!}}[\zeta (s+2)-1]\\&-{\frac {s(s+1)(s+2)}{3!}}\sum _{n=1}^{\infty }\int _{0}^{1}{\frac {t^{3}dt}{(n+t)^{s+3}}}\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Mellin-type_integrals">Mellin-type integrals</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=18" title="Edit section: Mellin-type integrals">edit</a>]</div> The <a href="/wiki/Mellin_transform" title="Mellin transform">Mellin transform</a> of a function f(x) is defined as<a href="#cite_note-32">[32]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }f(x)x^{s}\,{\frac {\mathrm {d} x}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }f(x)x^{s}\,{\frac {\mathrm {d} x}{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/344cfac0b7930a589e58108fca36373f17ff49ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.323ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }f(x)x^{s}\,{\frac {\mathrm {d} x}{x}}}"></dd></dl> in the region where the integral is defined. There are various expressions for the zeta function as Mellin transform-like integrals. If the real part of s is greater than one, we have <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (s)\zeta (s)=\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\,\mathrm {d} x\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (s)\zeta (s)=\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\,\mathrm {d} x\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcc429c7ed111fc5e6cf297033c2400c755b0ecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.599ex; height:6.176ex;" alt="{\displaystyle \Gamma (s)\zeta (s)=\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\,\mathrm {d} x\quad }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \Gamma (s)\zeta (s)={\frac {1}{2s}}\int _{0}^{\infty }{\frac {x^{s}}{\cosh(x)-1}}\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>s</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mrow> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \Gamma (s)\zeta (s)={\frac {1}{2s}}\int _{0}^{\infty }{\frac {x^{s}}{\cosh(x)-1}}\,\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa5c83e1676b0ef3aca0598b9e9818c60fe660f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.362ex; height:6.176ex;" alt="{\displaystyle \quad \Gamma (s)\zeta (s)={\frac {1}{2s}}\int _{0}^{\infty }{\frac {x^{s}}{\cosh(x)-1}}\,\mathrm {d} x}">,</dd></dl> where Γ denotes the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>. By modifying the <a href="/wiki/Contour_integration" title="Contour integration">contour</a>, Riemann showed that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\sin(\pi s)\Gamma (s)\zeta (s)=i\oint _{H}{\frac {(-x)^{s-1}}{e^{x}-1}}\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>s</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\sin(\pi s)\Gamma (s)\zeta (s)=i\oint _{H}{\frac {(-x)^{s-1}}{e^{x}-1}}\,\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12e9b4961a587907e5dfb7f3a038dc4a45ad5750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.54ex; height:6.343ex;" alt="{\displaystyle 2\sin(\pi s)\Gamma (s)\zeta (s)=i\oint _{H}{\frac {(-x)^{s-1}}{e^{x}-1}}\,\mathrm {d} x}"></dd></dl> for all s<a href="#cite_note-33">[33]</a> (where H denotes the <a href="/wiki/Hankel_contour" title="Hankel contour">Hankel contour</a>). We can also find expressions which relate to prime numbers and the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a>. If π(x) is the <a href="/wiki/Prime-counting_function" title="Prime-counting function">prime-counting function</a>, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \zeta (s)=s\int _{0}^{\infty }{\frac {\pi (x)}{x(x^{s}-1)}}\,\mathrm {d} x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \zeta (s)=s\int _{0}^{\infty }{\frac {\pi (x)}{x(x^{s}-1)}}\,\mathrm {d} x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/074401bd612660c0cbcd7e1be8b236cc6c0b5cfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.591ex; height:6.509ex;" alt="{\displaystyle \ln \zeta (s)=s\int _{0}^{\infty }{\frac {\pi (x)}{x(x^{s}-1)}}\,\mathrm {d} x,}"></dd></dl> for values with Re(s) > 1. A similar Mellin transform involves the Riemann function J(x), which counts prime powers pn with a weight of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/n⁠, so that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(x)=\sum {\frac {\pi \left(x^{\frac {1}{n}}\right)}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J(x)=\sum {\frac {\pi \left(x^{\frac {1}{n}}\right)}{n}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70582fd1083256a6afe4721a587ccd6a07e2696c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.627ex; height:7.509ex;" alt="{\displaystyle J(x)=\sum {\frac {\pi \left(x^{\frac {1}{n}}\right)}{n}}.}"></dd></dl> Now <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \zeta (s)=s\int _{0}^{\infty }J(x)x^{-s-1}\,\mathrm {d} x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>J</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \zeta (s)=s\int _{0}^{\infty }J(x)x^{-s-1}\,\mathrm {d} x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8deff64d533ad55a038ffc9363dcc032f6a104a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.603ex; height:5.843ex;" alt="{\displaystyle \ln \zeta (s)=s\int _{0}^{\infty }J(x)x^{-s-1}\,\mathrm {d} x.}"></dd></dl> These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's <a href="/wiki/Prime-counting_function" title="Prime-counting function">prime-counting function</a> is easier to work with, and π(x) can be recovered from it by <a href="/wiki/M%C3%B6bius_inversion_formula" title="Möbius inversion formula">Möbius inversion</a>. <div class="mw-heading mw-heading3"><h3 id="Theta_functions">Theta functions</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=19" title="Edit section: Theta functions">edit</a>]</div> The Riemann zeta function can be given by a Mellin transform<a href="#cite_note-34">[34]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)\zeta (s)=\int _{0}^{\infty }{\bigl (}\theta (it)-1{\bigr )}t^{{\frac {s}{2}}-1}\,\mathrm {d} t,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)\zeta (s)=\int _{0}^{\infty }{\bigl (}\theta (it)-1{\bigr )}t^{{\frac {s}{2}}-1}\,\mathrm {d} t,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77a56045601292ab75636e75b68c025c72ab6d60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.848ex; height:5.843ex;" alt="{\displaystyle 2\pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)\zeta (s)=\int _{0}^{\infty }{\bigl (}\theta (it)-1{\bigr )}t^{{\frac {s}{2}}-1}\,\mathrm {d} t,}"></dd></dl> in terms of <a href="/wiki/Theta_function" title="Theta function">Jacobi's theta function</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta (\tau )=\sum _{n=-\infty }^{\infty }e^{\pi in^{2}\tau }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mi>i</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>τ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta (\tau )=\sum _{n=-\infty }^{\infty }e^{\pi in^{2}\tau }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ccf80b5b6ef1916a9af8b64dac367a893c0042f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.913ex; height:6.843ex;" alt="{\displaystyle \theta (\tau )=\sum _{n=-\infty }^{\infty }e^{\pi in^{2}\tau }.}"></dd></dl> However, this integral only converges if the real part of s is greater than 1, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all s except 0 and 1: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)\zeta (s)={\frac {1}{s-1}}-{\frac {1}{s}}+{\frac {1}{2}}\int _{0}^{1}\left(\theta (it)-t^{-{\frac {1}{2}}}\right)t^{{\frac {s}{2}}-1}\,\mathrm {d} t+{\frac {1}{2}}\int _{1}^{\infty }{\bigl (}\theta (it)-1{\bigr )}t^{{\frac {s}{2}}-1}\,\mathrm {d} t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)\zeta (s)={\frac {1}{s-1}}-{\frac {1}{s}}+{\frac {1}{2}}\int _{0}^{1}\left(\theta (it)-t^{-{\frac {1}{2}}}\right)t^{{\frac {s}{2}}-1}\,\mathrm {d} t+{\frac {1}{2}}\int _{1}^{\infty }{\bigl (}\theta (it)-1{\bigr )}t^{{\frac {s}{2}}-1}\,\mathrm {d} t.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1142d3bb76ef71a97b927db0a3af93724393abe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:89.051ex; height:6.343ex;" alt="{\displaystyle \pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)\zeta (s)={\frac {1}{s-1}}-{\frac {1}{s}}+{\frac {1}{2}}\int _{0}^{1}\left(\theta (it)-t^{-{\frac {1}{2}}}\right)t^{{\frac {s}{2}}-1}\,\mathrm {d} t+{\frac {1}{2}}\int _{1}^{\infty }{\bigl (}\theta (it)-1{\bigr )}t^{{\frac {s}{2}}-1}\,\mathrm {d} t.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Laurent_series">Laurent series</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=20" title="Edit section: Laurent series">edit</a>]</div> The Riemann zeta function is <a href="/wiki/Meromorphic" class="mw-redirect" title="Meromorphic">meromorphic</a> with a single <a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">pole</a> of order one at s = 1. It can therefore be expanded as a <a href="/wiki/Laurent_series" title="Laurent series">Laurent series</a> about s = 1; the series development is then<a href="#cite_note-35">[35]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)={\frac {1}{s-1}}+\sum _{n=0}^{\infty }{\frac {\gamma _{n}}{n!}}(1-s)^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>s</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)={\frac {1}{s-1}}+\sum _{n=0}^{\infty }{\frac {\gamma _{n}}{n!}}(1-s)^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31fe1655e279add165b580ced26b44cb0a5782f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.632ex; height:6.843ex;" alt="{\displaystyle \zeta (s)={\frac {1}{s-1}}+\sum _{n=0}^{\infty }{\frac {\gamma _{n}}{n!}}(1-s)^{n}.}"></dd></dl> The constants γn here are called the <a href="/wiki/Stieltjes_constants" title="Stieltjes constants">Stieltjes constants</a> and can be defined by the <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">limit</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{n}=\lim _{m\rightarrow \infty }{\left(\left(\sum _{k=1}^{m}{\frac {(\ln k)^{n}}{k}}\right)-{\frac {(\ln m)^{n+1}}{n+1}}\right)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>k</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mi>k</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>m</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{n}=\lim _{m\rightarrow \infty }{\left(\left(\sum _{k=1}^{m}{\frac {(\ln k)^{n}}{k}}\right)-{\frac {(\ln m)^{n+1}}{n+1}}\right)}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5b36ba52ffe381add0f832e92aa3b02e25ed712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.962ex; height:7.509ex;" alt="{\displaystyle \gamma _{n}=\lim _{m\rightarrow \infty }{\left(\left(\sum _{k=1}^{m}{\frac {(\ln k)^{n}}{k}}\right)-{\frac {(\ln m)^{n+1}}{n+1}}\right)}.}"></dd></dl> The constant term γ0 is the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>. <div class="mw-heading mw-heading3"><h3 id="Integral">Integral</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=21" title="Edit section: Integral">edit</a>]</div> For all s ∈ ℂ, s ≠ 1, the integral relation (cf. <a href="/wiki/Abel%E2%80%93Plana_formula" title="Abel–Plana formula">Abel–Plana formula</a>) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \zeta (s)\ =\ {\frac {1}{\ s-1\ }}+{\frac {\ 1\ }{2}}+2\int _{0}^{\infty }{\frac {\sin(\ s\ \arctan t\ )}{\ \left(1+t^{2}\right)^{s/2}\left(e^{2\pi t}-1\right)\ }}\ \operatorname {d} t\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>1</mn> <mtext> </mtext> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mtext> </mtext> <mi>s</mi> <mtext> </mtext> <mi>arctan</mi> <mo>⁡</mo> <mi>t</mi> <mtext> </mtext> <mo stretchy="false">)</mo> </mrow> <mrow> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>t</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \zeta (s)\ =\ {\frac {1}{\ s-1\ }}+{\frac {\ 1\ }{2}}+2\int _{0}^{\infty }{\frac {\sin(\ s\ \arctan t\ )}{\ \left(1+t^{2}\right)^{s/2}\left(e^{2\pi t}-1\right)\ }}\ \operatorname {d} t\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d689e777b0294c32ef6e625daf6cee795228984" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:57.721ex; height:7.009ex;" alt="{\displaystyle \ \zeta (s)\ =\ {\frac {1}{\ s-1\ }}+{\frac {\ 1\ }{2}}+2\int _{0}^{\infty }{\frac {\sin(\ s\ \arctan t\ )}{\ \left(1+t^{2}\right)^{s/2}\left(e^{2\pi t}-1\right)\ }}\ \operatorname {d} t\ }"></dd></dl> holds true, which may be used for a numerical evaluation of the zeta function. <div class="mw-heading mw-heading3"><h3 id="Rising_factorial">Rising factorial</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=22" title="Edit section: Rising factorial">edit</a>]</div> Another series development using the <a href="/wiki/Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">rising factorial</a> valid for the entire complex plane is <a href="#cite_note-Knopp-31">[31]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)={\frac {s}{s-1}}-\sum _{n=1}^{\infty }{\bigl (}\zeta (s+n)-1{\bigr )}{\frac {s(s+1)\cdots (s+n-1)}{(n+1)!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)={\frac {s}{s-1}}-\sum _{n=1}^{\infty }{\bigl (}\zeta (s+n)-1{\bigr )}{\frac {s(s+1)\cdots (s+n-1)}{(n+1)!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aef05d982557fc2f960946305d17f1c2ee31f48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.079ex; height:6.843ex;" alt="{\displaystyle \zeta (s)={\frac {s}{s-1}}-\sum _{n=1}^{\infty }{\bigl (}\zeta (s+n)-1{\bigr )}{\frac {s(s+1)\cdots (s+n-1)}{(n+1)!}}.}"></dd></dl> This can be used recursively to extend the Dirichlet series definition to all complex numbers. The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the <a href="/wiki/Gauss%E2%80%93Kuzmin%E2%80%93Wirsing_operator" title="Gauss–Kuzmin–Wirsing operator">Gauss–Kuzmin–Wirsing operator</a> acting on xs − 1; that context gives rise to a series expansion in terms of the <a href="/wiki/Falling_factorial" class="mw-redirect" title="Falling factorial">falling factorial</a>.<a href="#cite_note-36">[36]</a> <div class="mw-heading mw-heading3"><h3 id="Hadamard_product">Hadamard product</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=23" title="Edit section: Hadamard product">edit</a>]</div> On the basis of <a href="/wiki/Weierstrass_factorization_theorem" title="Weierstrass factorization theorem">Weierstrass's factorization theorem</a>, <a href="/wiki/Hadamard" class="mw-redirect" title="Hadamard">Hadamard</a> gave the <a href="/wiki/Infinite_product" title="Infinite product">infinite product</a> expansion <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)={\frac {e^{\left(\log(2\pi )-1-{\frac {\gamma }{2}}\right)s}}{2(s-1)\Gamma \left(1+{\frac {s}{2}}\right)}}\prod _{\rho }\left(1-{\frac {s}{\rho }}\right)e^{\frac {s}{\rho }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>s</mi> </mrow> </msup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>ρ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>ρ</mi> </mfrac> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)={\frac {e^{\left(\log(2\pi )-1-{\frac {\gamma }{2}}\right)s}}{2(s-1)\Gamma \left(1+{\frac {s}{2}}\right)}}\prod _{\rho }\left(1-{\frac {s}{\rho }}\right)e^{\frac {s}{\rho }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d40cc4315011346c4af33c07cdfef83c509df1f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:42.707ex; height:8.343ex;" alt="{\displaystyle \zeta (s)={\frac {e^{\left(\log(2\pi )-1-{\frac {\gamma }{2}}\right)s}}{2(s-1)\Gamma \left(1+{\frac {s}{2}}\right)}}\prod _{\rho }\left(1-{\frac {s}{\rho }}\right)e^{\frac {s}{\rho }},}"></dd></dl> where the product is over the non-trivial zeros ρ of ζ and the letter γ again denotes the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>. A simpler <a href="/wiki/Infinite_product" title="Infinite product">infinite product</a> expansion is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)=\pi ^{\frac {s}{2}}{\frac {\prod _{\rho }\left(1-{\frac {s}{\rho }}\right)}{2(s-1)\Gamma \left(1+{\frac {s}{2}}\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>ρ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)=\pi ^{\frac {s}{2}}{\frac {\prod _{\rho }\left(1-{\frac {s}{\rho }}\right)}{2(s-1)\Gamma \left(1+{\frac {s}{2}}\right)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/308e0c0b2db471d9d4eaf38da2b1ab6e4898b297" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.342ex; height:8.843ex;" alt="{\displaystyle \zeta (s)=\pi ^{\frac {s}{2}}{\frac {\prod _{\rho }\left(1-{\frac {s}{\rho }}\right)}{2(s-1)\Gamma \left(1+{\frac {s}{2}}\right)}}.}"></dd></dl> This form clearly displays the simple pole at s = 1, the trivial zeros at −2, −4, ... due to the gamma function term in the denominator, and the non-trivial zeros at s = ρ. (To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i.e. the factors for a pair of zeros of the form ρ and 1 − ρ should be combined.) <div class="mw-heading mw-heading3"><h3 id="Globally_convergent_series">Globally convergent series</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=24" title="Edit section: Globally convergent series">edit</a>]</div> A globally convergent series for the zeta function, valid for all complex numbers s except s = 1 + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2πi/ln 2⁠n for some integer n, was conjectured by <a href="/wiki/Konrad_Knopp" title="Konrad Knopp">Konrad Knopp</a> in 1926 <a href="#cite_note-blag2018-37">[37]</a> and proven by <a href="/wiki/Helmut_Hasse" title="Helmut Hasse">Helmut Hasse</a> in 1930<a href="#cite_note-Hasse1930-38">[38]</a> (cf. <a href="/wiki/Euler_summation" title="Euler summation">Euler summation</a>): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)={\frac {1}{1-2^{1-s}}}\sum _{n=0}^{\infty }{\frac {1}{2^{n+1}}}\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{(k+1)^{s}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)={\frac {1}{1-2^{1-s}}}\sum _{n=0}^{\infty }{\frac {1}{2^{n+1}}}\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{(k+1)^{s}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e8083dd600ae6b8f056a638d4f140b47abc8011" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.1ex; height:7.176ex;" alt="{\displaystyle \zeta (s)={\frac {1}{1-2^{1-s}}}\sum _{n=0}^{\infty }{\frac {1}{2^{n+1}}}\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{(k+1)^{s}}}.}"></dd></dl> The series appeared in an appendix to Hasse's paper, and was published for the second time by Jonathan Sondow in 1994.<a href="#cite_note-39">[39]</a> Hasse also proved the globally converging series <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{(k+1)^{s-1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{(k+1)^{s-1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02a8e399f217daa6ccbf79ca537540bfb367e692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.294ex; height:7.176ex;" alt="{\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{(k+1)^{s-1}}}}"></dd></dl> in the same publication.<a href="#cite_note-Hasse1930-38">[38]</a> Research by Iaroslav Blagouchine<a href="#cite_note-40">[40]</a><a href="#cite_note-blag2018-37">[37]</a> has found that a similar, equivalent series was published by <a href="/wiki/Joseph_Ser" title="Joseph Ser">Joseph Ser</a> in 1926.<a href="#cite_note-41">[41]</a> In 1997 K. Maślanka gave another globally convergent (except s = 1) series for the Riemann zeta function: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{k=0}^{\infty }{\biggl (}\prod _{i=1}^{k}(i-{\frac {s}{2}}){\biggl )}{\frac {A_{k}}{k!}}={\frac {1}{s-1}}\sum _{k=0}^{\infty }{\biggl (}1-{\frac {s}{2}}{\biggl )}_{k}{\frac {A_{k}}{k!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>i</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{k=0}^{\infty }{\biggl (}\prod _{i=1}^{k}(i-{\frac {s}{2}}){\biggl )}{\frac {A_{k}}{k!}}={\frac {1}{s-1}}\sum _{k=0}^{\infty }{\biggl (}1-{\frac {s}{2}}{\biggl )}_{k}{\frac {A_{k}}{k!}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/471084c94afec842efa2eececae28e012df5a6f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.385ex; height:7.509ex;" alt="{\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{k=0}^{\infty }{\biggl (}\prod _{i=1}^{k}(i-{\frac {s}{2}}){\biggl )}{\frac {A_{k}}{k!}}={\frac {1}{s-1}}\sum _{k=0}^{\infty }{\biggl (}1-{\frac {s}{2}}{\biggl )}_{k}{\frac {A_{k}}{k!}}}"></dd></dl> where real coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72095229db907e86eb4343cb4736429fcc56507d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.832ex; height:2.509ex;" alt="{\displaystyle A_{k}}"> are given by: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{k}=\sum _{j=0}^{k}(-1)^{j}{\binom {k}{j}}(2j+1)\zeta (2j+2)=\sum _{j=0}^{k}{\binom {k}{j}}{\frac {B_{2j+2}\pi ^{2j+2}}{\left(2\right)_{j}\left({\frac {1}{2}}\right)_{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{k}=\sum _{j=0}^{k}(-1)^{j}{\binom {k}{j}}(2j+1)\zeta (2j+2)=\sum _{j=0}^{k}{\binom {k}{j}}{\frac {B_{2j+2}\pi ^{2j+2}}{\left(2\right)_{j}\left({\frac {1}{2}}\right)_{j}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bab78583646f7a680093d11e692b9bab82595af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:59.639ex; height:8.176ex;" alt="{\displaystyle A_{k}=\sum _{j=0}^{k}(-1)^{j}{\binom {k}{j}}(2j+1)\zeta (2j+2)=\sum _{j=0}^{k}{\binom {k}{j}}{\frac {B_{2j+2}\pi ^{2j+2}}{\left(2\right)_{j}\left({\frac {1}{2}}\right)_{j}}}}"></dd></dl> Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}"> are the Bernoulli numbers and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x)_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cd9ea305ea82c2adc62bee97f7178226f2940bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.228ex; height:2.843ex;" alt="{\displaystyle (x)_{k}}"> denotes the Pochhammer symbol.<a href="#cite_note-42">[42]</a><a href="#cite_note-43">[43]</a> Note that this representation of the zeta function is essentially an interpolation with nodes, where the nodes are points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=2,4,6,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=2,4,6,\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61cb4a2621b1e87066f876921729538d349b9f82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.501ex; height:2.509ex;" alt="{\displaystyle s=2,4,6,\ldots }">, i.e. exactly those where the zeta values are precisely known, as Euler showed. An elegant and very short proof of this representation of the zeta function, based on Carlson's theorem, was presented by Philippe Flajolet in 2006.<a href="#cite_note-44">[44]</a> The asymptotic behavior of the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72095229db907e86eb4343cb4736429fcc56507d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.832ex; height:2.509ex;" alt="{\displaystyle A_{k}}"> is rather curious: for growing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> values, we observe regular oscillations with a nearly exponentially decreasing amplitude and slowly decreasing frequency (roughly as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{-2/3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{-2/3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82a486efbb6c10c3faa9da3a5649ce1ee77f3193" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.188ex; height:2.843ex;" alt="{\displaystyle k^{-2/3}}">). Using the saddle point method, we can show that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{k}\sim {\frac {4\pi ^{3/2}}{\sqrt {3\kappa }}}\exp {\biggl (}-{\frac {3\kappa }{2}}+{\frac {\pi ^{2}}{4\kappa }}{\biggl )}\cos {\biggl (}{\frac {4\pi }{3}}-{\frac {3{\sqrt {3}}\kappa }{2}}+{\frac {{\sqrt {3}}\pi ^{2}}{4\kappa }}{\biggl )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> <msqrt> <mn>3</mn> <mi>κ</mi> </msqrt> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>κ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>κ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mi>κ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mi>κ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{k}\sim {\frac {4\pi ^{3/2}}{\sqrt {3\kappa }}}\exp {\biggl (}-{\frac {3\kappa }{2}}+{\frac {\pi ^{2}}{4\kappa }}{\biggl )}\cos {\biggl (}{\frac {4\pi }{3}}-{\frac {3{\sqrt {3}}\kappa }{2}}+{\frac {{\sqrt {3}}\pi ^{2}}{4\kappa }}{\biggl )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7df591437c6cb6e355468accefa456bc6508d42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:61.142ex; height:6.843ex;" alt="{\displaystyle A_{k}\sim {\frac {4\pi ^{3/2}}{\sqrt {3\kappa }}}\exp {\biggl (}-{\frac {3\kappa }{2}}+{\frac {\pi ^{2}}{4\kappa }}{\biggl )}\cos {\biggl (}{\frac {4\pi }{3}}-{\frac {3{\sqrt {3}}\kappa }{2}}+{\frac {{\sqrt {3}}\pi ^{2}}{4\kappa }}{\biggl )}}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"> stands for: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa :={\sqrt[{3}]{\pi ^{2}k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa :={\sqrt[{3}]{\pi ^{2}k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0339cf277c68851802c4df552e62d4b84e031820" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.008ex; height:3.343ex;" alt="{\displaystyle \kappa :={\sqrt[{3}]{\pi ^{2}k}}}"></dd></dl> (see <a href="#cite_note-45">[45]</a> for details). On the basis of this representation, in 2003 Luis Báez-Duarte provided a new criterion for the Riemann hypothesis.<a href="#cite_note-46">[46]</a><a href="#cite_note-47">[47]</a><a href="#cite_note-48">[48]</a> Namely, if we define the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2f8052630e67b00d04e3487e1d68ed7070470b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.096ex; height:2.009ex;" alt="{\displaystyle c_{k}}"> as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{k}:=\sum _{j=0}^{k}(-1)^{j}{\binom {k}{j}}{\frac {1}{\zeta (2j+2)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>:=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{k}:=\sum _{j=0}^{k}(-1)^{j}{\binom {k}{j}}{\frac {1}{\zeta (2j+2)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3e698b5ea1e24ae7ddb8ddc8bdc42889de2d067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:29.382ex; height:7.676ex;" alt="{\displaystyle c_{k}:=\sum _{j=0}^{k}(-1)^{j}{\binom {k}{j}}{\frac {1}{\zeta (2j+2)}}}"></dd></dl> then the Riemann hypothesis is equivalent to <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{k}={\mathcal {O}}{\biggl (}k^{-3/4+\varepsilon }{\biggl )}\qquad (\forall \varepsilon >0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{k}={\mathcal {O}}{\biggl (}k^{-3/4+\varepsilon }{\biggl )}\qquad (\forall \varepsilon >0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e75c259ad8da6ee3e805b45ede7c7deceb4762" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.789ex; height:6.176ex;" alt="{\displaystyle c_{k}={\mathcal {O}}{\biggl (}k^{-3/4+\varepsilon }{\biggl )}\qquad (\forall \varepsilon >0)}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Rapidly_convergent_series">Rapidly convergent series</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=25" title="Edit section: Rapidly convergent series">edit</a>]</div> <a href="/wiki/Peter_Borwein" title="Peter Borwein">Peter Borwein</a> developed an algorithm that applies <a href="/wiki/Chebyshev_polynomial" class="mw-redirect" title="Chebyshev polynomial">Chebyshev polynomials</a> to the <a href="/wiki/Dirichlet_eta_function" title="Dirichlet eta function">Dirichlet eta function</a> to produce a <a href="/wiki/Dirichlet_eta_function#Borwein's_method" title="Dirichlet eta function">very rapidly convergent series suitable for high precision numerical calculations</a>.<a href="#cite_note-49">[49]</a> <div class="mw-heading mw-heading3"><h3 id="Series_representation_at_positive_integers_via_the_primorial">Series representation at positive integers via the primorial</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=26" title="Edit section: Series representation at positive integers via the primorial">edit</a>]</div> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (k)={\frac {2^{k}}{2^{k}-1}}+\sum _{r=2}^{\infty }{\frac {(p_{r-1}\#)^{k}}{J_{k}(p_{r}\#)}}\qquad k=2,3,\ldots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi mathvariant="normal">#</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mi mathvariant="normal">#</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (k)={\frac {2^{k}}{2^{k}-1}}+\sum _{r=2}^{\infty }{\frac {(p_{r-1}\#)^{k}}{J_{k}(p_{r}\#)}}\qquad k=2,3,\ldots .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03b43a21967fc389206d243734a9dbefb3514f49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.905ex; height:7.009ex;" alt="{\displaystyle \zeta (k)={\frac {2^{k}}{2^{k}-1}}+\sum _{r=2}^{\infty }{\frac {(p_{r-1}\#)^{k}}{J_{k}(p_{r}\#)}}\qquad k=2,3,\ldots .}"></dd></dl> Here pn# is the <a href="/wiki/Primorial" title="Primorial">primorial</a> sequence and Jk is <a href="/wiki/Jordan%27s_totient_function" title="Jordan's totient function">Jordan's totient function</a>.<a href="#cite_note-50">[50]</a> <div class="mw-heading mw-heading3"><h3 id="Series_representation_by_the_incomplete_poly-Bernoulli_numbers">Series representation by the incomplete poly-Bernoulli numbers</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=27" title="Edit section: Series representation by the incomplete poly-Bernoulli numbers">edit</a>]</div> The function ζ can be represented, for Re(s) > 1, by the infinite series <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)=\sum _{n=0}^{\infty }B_{n,\geq 2}^{(s)}{\frac {(W_{k}(-1))^{n}}{n!}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mo>≥</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)=\sum _{n=0}^{\infty }B_{n,\geq 2}^{(s)}{\frac {(W_{k}(-1))^{n}}{n!}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45b6c460ffb2250ec4fddb5e2977d0cdb2220b11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.949ex; height:6.843ex;" alt="{\displaystyle \zeta (s)=\sum _{n=0}^{\infty }B_{n,\geq 2}^{(s)}{\frac {(W_{k}(-1))^{n}}{n!}},}"></dd></dl> where k ∈ {−1, 0}, Wk is the kth branch of the <a href="/wiki/Lambert_W_function" title="Lambert W function">Lambert W-function</a>, and B(μ) n, ≥2 is an incomplete poly-Bernoulli number.<a href="#cite_note-51">[51]</a> <div class="mw-heading mw-heading3"><h3 id="The_Mellin_transform_of_the_Engel_map">The Mellin transform of the Engel map</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=28" title="Edit section: The Mellin transform of the Engel map">edit</a>]</div> The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=x\left(1+\left\lfloor x^{-1}\right\rfloor \right)-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <mo>⌊</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⌋</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=x\left(1+\left\lfloor x^{-1}\right\rfloor \right)-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f72dfffefdae0069068211e2e6ffb60f46c153eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.063ex; height:3.343ex;" alt="{\displaystyle g(x)=x\left(1+\left\lfloor x^{-1}\right\rfloor \right)-1}"> is iterated to find the coefficients appearing in <a href="/wiki/Engel_expansion" title="Engel expansion">Engel expansions</a>.<a href="#cite_note-52">[52]</a> The <a href="/wiki/Mellin_transform" title="Mellin transform">Mellin transform</a> of the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}"> is related to the Riemann zeta function by the formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{0}^{1}g(x)x^{s-1}\,dx&=\sum _{n=1}^{\infty }\int _{\frac {1}{n+1}}^{\frac {1}{n}}(x(n+1)-1)x^{s-1}\,dx\\[6pt]&=\sum _{n=1}^{\infty }{\frac {n^{-s}(s-1)+(n+1)^{-s-1}(n^{2}+2n+1)+n^{-s-1}s-n^{1-s}}{(s+1)s(n+1)}}\\[6pt]&={\frac {\zeta (s+1)}{s+1}}-{\frac {1}{s(s+1)}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>s</mi> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{0}^{1}g(x)x^{s-1}\,dx&=\sum _{n=1}^{\infty }\int _{\frac {1}{n+1}}^{\frac {1}{n}}(x(n+1)-1)x^{s-1}\,dx\\[6pt]&=\sum _{n=1}^{\infty }{\frac {n^{-s}(s-1)+(n+1)^{-s-1}(n^{2}+2n+1)+n^{-s-1}s-n^{1-s}}{(s+1)s(n+1)}}\\[6pt]&={\frac {\zeta (s+1)}{s+1}}-{\frac {1}{s(s+1)}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e030b32f7b471b521e7cc74a30548917e1d5443" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.671ex; width:79.669ex; height:24.509ex;" alt="{\displaystyle {\begin{aligned}\int _{0}^{1}g(x)x^{s-1}\,dx&=\sum _{n=1}^{\infty }\int _{\frac {1}{n+1}}^{\frac {1}{n}}(x(n+1)-1)x^{s-1}\,dx\\[6pt]&=\sum _{n=1}^{\infty }{\frac {n^{-s}(s-1)+(n+1)^{-s-1}(n^{2}+2n+1)+n^{-s-1}s-n^{1-s}}{(s+1)s(n+1)}}\\[6pt]&={\frac {\zeta (s+1)}{s+1}}-{\frac {1}{s(s+1)}}\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Thue-Morse_sequence">Thue-Morse sequence</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=29" title="Edit section: Thue-Morse sequence">edit</a>]</div> Certain linear combinations of Dirichlet series whose coefficients are terms of the <a href="/wiki/Thue-Morse_sequence" class="mw-redirect" title="Thue-Morse sequence">Thue-Morse sequence</a> give rise to identities involving the Riemann Zeta function (Tóth, 2022 <a href="#cite_note-53">[53]</a>). For instance: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {5t_{n-1}+3t_{n}}{n^{2}}}&=4\zeta (2)={\frac {2\pi ^{2}}{3}},\\\sum _{n\geq 1}{\frac {9t_{n-1}+7t_{n}}{n^{3}}}&=8\zeta (3),\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>9</mn> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>7</mn> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>8</mn> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {5t_{n-1}+3t_{n}}{n^{2}}}&=4\zeta (2)={\frac {2\pi ^{2}}{3}},\\\sum _{n\geq 1}{\frac {9t_{n-1}+7t_{n}}{n^{3}}}&=8\zeta (3),\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c86839ac7944330fb8b82dbdd6c87ae80f99f6f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:33.172ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {5t_{n-1}+3t_{n}}{n^{2}}}&=4\zeta (2)={\frac {2\pi ^{2}}{3}},\\\sum _{n\geq 1}{\frac {9t_{n-1}+7t_{n}}{n^{3}}}&=8\zeta (3),\end{aligned}}}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t_{n})_{n\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t_{n})_{n\geq 0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14bd857801110e4b9e904aefe66b16d98536828" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.186ex; height:2.843ex;" alt="{\displaystyle (t_{n})_{n\geq 0}}"> is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{\rm {th}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{\rm {th}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d964c2d57fab47cb34cb1ef2b2c8d96f3e78e7e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.181ex; height:2.676ex;" alt="{\displaystyle n^{\rm {th}}}"> term of the Thue-Morse sequence. In fact, for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> with real part greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">, we have <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2^{s}+1)\sum _{n\geq 1}{\frac {t_{n-1}}{n^{s}}}+(2^{s}-1)\sum _{n\geq 1}{\frac {t_{n}}{n^{s}}}=2^{s}\zeta (s).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2^{s}+1)\sum _{n\geq 1}{\frac {t_{n-1}}{n^{s}}}+(2^{s}-1)\sum _{n\geq 1}{\frac {t_{n}}{n^{s}}}=2^{s}\zeta (s).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b587abde1267f39dd6d25ab6c5fafe217c0b629a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.189ex; height:6.509ex;" alt="{\displaystyle (2^{s}+1)\sum _{n\geq 1}{\frac {t_{n-1}}{n^{s}}}+(2^{s}-1)\sum _{n\geq 1}{\frac {t_{n}}{n^{s}}}=2^{s}\zeta (s).}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="In_nth_dimensions">In nth dimensions</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=30" title="Edit section: In nth dimensions">edit</a>]</div> The zeta function can also be represented as an nth amount of integrals: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (n)=\int _{\cdots }\iint _{[0,1]^{n}}{\frac {dz_{1}\,dz_{2}\ldots \,dz_{n}}{1-z_{1}\,z_{2}\ldots \,z_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⋯</mo> </mrow> </msub> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <mspace width="thinmathspace" /> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (n)=\int _{\cdots }\iint _{[0,1]^{n}}{\frac {dz_{1}\,dz_{2}\ldots \,dz_{n}}{1-z_{1}\,z_{2}\ldots \,z_{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fdbaeae8149f84d0316c3c4b15a0541e55ee79b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.035ex; height:6.176ex;" alt="{\displaystyle \zeta (n)=\int _{\cdots }\iint _{[0,1]^{n}}{\frac {dz_{1}\,dz_{2}\ldots \,dz_{n}}{1-z_{1}\,z_{2}\ldots \,z_{n}}}}"> and it only works for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} /\{1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} /\{1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adec1a218419dc35fec8b6503787800d3193030d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.563ex; height:2.843ex;" alt="{\displaystyle n\in \mathbb {N} /\{1\}}"> <div class="mw-heading mw-heading2"><h2 id="Numerical_algorithms">Numerical algorithms</h2>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=31" title="Edit section: Numerical algorithms">edit</a>]</div> A classical algorithm, in use prior to about 1930, proceeds by applying the <a href="/wiki/Euler-Maclaurin_formula" class="mw-redirect" title="Euler-Maclaurin formula">Euler-Maclaurin formula</a> to obtain, for n and m positive integers, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)=\sum _{j=1}^{n-1}j^{-s}+{\tfrac {1}{2}}n^{-s}+{\frac {n^{1-s}}{s-1}}+\sum _{k=1}^{m}T_{k,n}(s)+E_{m,n}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)=\sum _{j=1}^{n-1}j^{-s}+{\tfrac {1}{2}}n^{-s}+{\frac {n^{1-s}}{s-1}}+\sum _{k=1}^{m}T_{k,n}(s)+E_{m,n}(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcedbaa43360a1fd9aece6db06ec6c004c92791a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:54.966ex; height:7.676ex;" alt="{\displaystyle \zeta (s)=\sum _{j=1}^{n-1}j^{-s}+{\tfrac {1}{2}}n^{-s}+{\frac {n^{1-s}}{s-1}}+\sum _{k=1}^{m}T_{k,n}(s)+E_{m,n}(s)}"></dd></dl> where, letting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{2k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d01575253488226c50dd432a9a4c1992f2990159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.675ex; height:2.509ex;" alt="{\displaystyle B_{2k}}"> denote the indicated <a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli number</a>, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{k,n}(s)={\frac {B_{2k}}{(2k)!}}n^{1-s-2k}\prod _{j=0}^{2k-2}(s+j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mo>−</mo> <mn>2</mn> <mi>k</mi> </mrow> </msup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>2</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{k,n}(s)={\frac {B_{2k}}{(2k)!}}n^{1-s-2k}\prod _{j=0}^{2k-2}(s+j)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4311a4cac3a221ec90f3338811d50fdd9c318165" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.874ex; height:7.676ex;" alt="{\displaystyle T_{k,n}(s)={\frac {B_{2k}}{(2k)!}}n^{1-s-2k}\prod _{j=0}^{2k-2}(s+j)}"></dd></dl> and the error satisfies <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E_{m,n}(s)|<\left|{\frac {s+2m+1}{\sigma +2m+1}}T_{m+1,n}(s)\right|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo>+</mo> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>σ</mi> <mo>+</mo> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |E_{m,n}(s)|<\left|{\frac {s+2m+1}{\sigma +2m+1}}T_{m+1,n}(s)\right|,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/361965b8b53ec77bef0c2bd826a3222285717ac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:36.142ex; height:5.509ex;" alt="{\displaystyle |E_{m,n}(s)|<\left|{\frac {s+2m+1}{\sigma +2m+1}}T_{m+1,n}(s)\right|,}"></dd></dl> with σ = Re(s).<a href="#cite_note-54">[54]</a> A modern numerical algorithm is the <a href="/wiki/Odlyzko%E2%80%93Sch%C3%B6nhage_algorithm" title="Odlyzko–Schönhage algorithm">Odlyzko–Schönhage algorithm</a>. <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=32" title="Edit section: Applications">edit</a>]</div> The zeta function occurs in applied <a href="/wiki/Statistics" title="Statistics">statistics</a> including <a href="/wiki/Zipf%27s_law" title="Zipf's law">Zipf's law</a>, <a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="Zipf–Mandelbrot law">Zipf–Mandelbrot law</a>, and <a href="/wiki/Lotka%27s_law" title="Lotka's law">Lotka's law</a>. <a href="/wiki/Zeta_function_regularization" title="Zeta function regularization">Zeta function regularization</a> is used as one possible means of <a href="/wiki/Regularization_(physics)" title="Regularization (physics)">regularization</a> of <a href="/wiki/Divergent_series" title="Divergent series">divergent series</a> and <a href="/wiki/Divergent_integral" class="mw-redirect" title="Divergent integral">divergent integrals</a> in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. In one notable example, the Riemann zeta function shows up explicitly in one method of calculating the <a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a>. The zeta function is also useful for the analysis of <a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a>.<a href="#cite_note-55">[55]</a> <div class="mw-heading mw-heading3"><h3 id="Musical_tuning">Musical tuning</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=33" title="Edit section: Musical tuning">edit</a>]</div> In the theory of <a href="/wiki/Musical_tuning" title="Musical tuning">musical tunings</a>, the zeta function can be used to find <a href="/wiki/Equal_temperament" title="Equal temperament">equal divisions of the octave</a> (EDOs) that closely approximate the intervals of the <a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">harmonic series</a>. For increasing values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/592bced0c39b10fc90e74c6a66223abfbfb029de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.358ex; height:2.176ex;" alt="{\displaystyle t\in \mathbb {R} }">, the value of <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\vert \zeta \left({\frac {1}{2}}+{\frac {2\pi {i}}{\ln {(2)}}}t\right)\right\vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>ζ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </mrow> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\vert \zeta \left({\frac {1}{2}}+{\frac {2\pi {i}}{\ln {(2)}}}t\right)\right\vert }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6990e640855b94053af469b0ea80b63dce441a7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.01ex; height:6.509ex;" alt="{\displaystyle \left\vert \zeta \left({\frac {1}{2}}+{\frac {2\pi {i}}{\ln {(2)}}}t\right)\right\vert }"></dd></dl> peaks near integers that correspond to such EDOs.<a href="#cite_note-56">[56]</a> Examples include popular choices such as 12, 19, and 53.<a href="#cite_note-57">[57]</a> <div class="mw-heading mw-heading3"><h3 id="Infinite_series">Infinite series</h3>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=34" title="Edit section: Infinite series">edit</a>]</div> The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.<a href="#cite_note-58">[58]</a> <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=2}^{\infty }{\bigl (}\zeta (n)-1{\bigr )}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=2}^{\infty }{\bigl (}\zeta (n)-1{\bigr )}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8fae8dd436dacd8a9f6264c8de78db55b70bbc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.435ex; height:6.843ex;" alt="{\displaystyle \sum _{n=2}^{\infty }{\bigl (}\zeta (n)-1{\bigr )}=1}"></li></ul> In fact the even and odd terms give the two sums <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (2n)-1{\bigr )}={\frac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (2n)-1{\bigr )}={\frac {3}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59de0623a80e18e1b0a9f081f9c7c86175c29146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.433ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (2n)-1{\bigr )}={\frac {3}{4}}}"></li></ul> and <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (2n+1)-1{\bigr )}={\frac {1}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (2n+1)-1{\bigr )}={\frac {1}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a42c47494e3bfe4d5be6f10ea3c9131b76a2d7b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.436ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (2n+1)-1{\bigr )}={\frac {1}{4}}}"></li></ul> Parametrized versions of the above sums are given by <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }(\zeta (2n)-1)\,t^{2n}={\frac {t^{2}}{t^{2}-1}}+{\frac {1}{2}}\left(1-\pi t\cot(t\pi )\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>π</mi> <mi>t</mi> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }(\zeta (2n)-1)\,t^{2n}={\frac {t^{2}}{t^{2}-1}}+{\frac {1}{2}}\left(1-\pi t\cot(t\pi )\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b83308eb216a26952fa784532b72e07f1275fabc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.405ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }(\zeta (2n)-1)\,t^{2n}={\frac {t^{2}}{t^{2}-1}}+{\frac {1}{2}}\left(1-\pi t\cot(t\pi )\right)}"></li></ul> and <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }(\zeta (2n+1)-1)\,t^{2n}={\frac {t^{2}}{t^{2}-1}}-{\frac {1}{2}}\left(\psi ^{0}(t)+\psi ^{0}(-t)\right)-\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }(\zeta (2n+1)-1)\,t^{2n}={\frac {t^{2}}{t^{2}-1}}-{\frac {1}{2}}\left(\psi ^{0}(t)+\psi ^{0}(-t)\right)-\gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7836c087c881ce8782c966fbddf8d51e5a39cf71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.27ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }(\zeta (2n+1)-1)\,t^{2n}={\frac {t^{2}}{t^{2}-1}}-{\frac {1}{2}}\left(\psi ^{0}(t)+\psi ^{0}(-t)\right)-\gamma }"></li></ul> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |t|<2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |t|<2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61194a1f0eb313fc72639add2d635df5e396d354" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" alt="{\displaystyle |t|<2}"> and where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"> are the <a href="/wiki/Polygamma_function" title="Polygamma function">polygamma function</a> and <a href="/wiki/Euler%27s_constant" title="Euler's constant">Euler's constant</a>, respectively, as well as <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n}}\,t^{2n}=\log \left({\dfrac {1-t^{2}}{\operatorname {sinc} (\pi \,t)}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>sinc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mspace width="thinmathspace" /> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n}}\,t^{2n}=\log \left({\dfrac {1-t^{2}}{\operatorname {sinc} (\pi \,t)}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcb0b7b1eeeede755faf8bf3463a1eef71ff0957" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.893ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n}}\,t^{2n}=\log \left({\dfrac {1-t^{2}}{\operatorname {sinc} (\pi \,t)}}\right)}"></li></ul> all of which are continuous at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/970dea4a5f5ec5355c4cdd62f6396fbc8b1baaa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=1}">. Other sums include <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}=1-\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}=1-\gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a08241be47efa01f9a27c070380118fcffab37d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.244ex; height:6.843ex;" alt="{\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}=1-\gamma }"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n}}=\ln 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n}}=\ln 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18bf2bd526cb0082a7ab1a68e33a943783c8a1bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.63ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n}}=\ln 2}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}\left(\left({\tfrac {3}{2}}\right)^{n-1}-1\right)={\frac {1}{3}}\ln \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}\left(\left({\tfrac {3}{2}}\right)^{n-1}-1\right)={\frac {1}{3}}\ln \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5e6f5c48af4dba6bb20b1a012df33894bdcc153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.587ex; height:6.843ex;" alt="{\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}\left(\left({\tfrac {3}{2}}\right)^{n-1}-1\right)={\frac {1}{3}}\ln \pi }"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (4n)-1{\bigr )}={\frac {7}{8}}-{\frac {\pi }{4}}\left({\frac {e^{2\pi }+1}{e^{2\pi }-1}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>8</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (4n)-1{\bigr )}={\frac {7}{8}}-{\frac {\pi }{4}}\left({\frac {e^{2\pi }+1}{e^{2\pi }-1}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/827ac299e8811d4cf0af4648b92e6519f909b12c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.203ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (4n)-1{\bigr )}={\frac {7}{8}}-{\frac {\pi }{4}}\left({\frac {e^{2\pi }+1}{e^{2\pi }-1}}\right).}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}\Im {\bigl (}(1+i)^{n}-1-i^{n}{\bigr )}={\frac {\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mi mathvariant="normal">ℑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}\Im {\bigl (}(1+i)^{n}-1-i^{n}{\bigr )}={\frac {\pi }{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd33b640d6c0bf1a7be6f2536f20096c24b7817" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.262ex; height:6.843ex;" alt="{\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}\Im {\bigl (}(1+i)^{n}-1-i^{n}{\bigr )}={\frac {\pi }{4}}}"></li></ul> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Im }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">ℑ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Im }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3e0312a4871a615cbdae168be102907f0a51e95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.288ex; height:2.176ex;" alt="{\displaystyle \Im }"> denotes the <a href="/wiki/Imaginary_part" class="mw-redirect" title="Imaginary part">imaginary part</a> of a complex number. Another interesting series that relates to the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> of the <a href="/wiki/Lemniscate_constant" title="Lemniscate constant">lemniscate constant</a> is the following <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=2}^{\infty }\left[{\frac {2(-1)^{n}\zeta (n)}{4^{n}n}}-{\frac {(-1)^{n}\zeta (n)}{2^{n}n}}\right]=\ln \left({\frac {\varpi }{2{\sqrt {2}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϖ</mi> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=2}^{\infty }\left[{\frac {2(-1)^{n}\zeta (n)}{4^{n}n}}-{\frac {(-1)^{n}\zeta (n)}{2^{n}n}}\right]=\ln \left({\frac {\varpi }{2{\sqrt {2}}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2458d76fa250bf8e6073c0983d369ad65337e9c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.023ex; height:6.843ex;" alt="{\displaystyle \sum _{n=2}^{\infty }\left[{\frac {2(-1)^{n}\zeta (n)}{4^{n}n}}-{\frac {(-1)^{n}\zeta (n)}{2^{n}n}}\right]=\ln \left({\frac {\varpi }{2{\sqrt {2}}}}\right)}"></li></ul> There are yet more formulas in the article <a href="/wiki/Harmonic_number#Relation_to_the_Riemann_zeta_function" title="Harmonic number">Harmonic number.</a> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=35" title="Edit section: Generalizations">edit</a>]</div> There are a number of related <a href="/wiki/Zeta_function" class="mw-redirect" title="Zeta function">zeta functions</a> that can be considered to be generalizations of the Riemann zeta function. These include the <a href="/wiki/Hurwitz_zeta_function" title="Hurwitz zeta function">Hurwitz zeta function</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s,q)=\sum _{k=0}^{\infty }{\frac {1}{(k+q)^{s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s,q)=\sum _{k=0}^{\infty }{\frac {1}{(k+q)^{s}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38163e2c24d461276aea2cb59ea13cbc7fceb28d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.709ex; height:7.009ex;" alt="{\displaystyle \zeta (s,q)=\sum _{k=0}^{\infty }{\frac {1}{(k+q)^{s}}}}"></dd></dl> (the convergent series representation was given by <a href="/wiki/Helmut_Hasse" title="Helmut Hasse">Helmut Hasse</a> in 1930,<a href="#cite_note-Hasse1930-38">[38]</a> cf. <a href="/wiki/Hurwitz_zeta_function" title="Hurwitz zeta function">Hurwitz zeta function</a>), which coincides with the Riemann zeta function when q = 1 (the lower limit of summation in the Hurwitz zeta function is 0, not 1), the <a href="/wiki/Dirichlet_L-function" title="Dirichlet L-function">Dirichlet L-functions</a> and the <a href="/wiki/Dedekind_zeta_function" title="Dedekind zeta function">Dedekind zeta function</a>. For other related functions see the articles <a href="/wiki/Zeta_function" class="mw-redirect" title="Zeta function">zeta function</a> and <a href="/wiki/L-function" title="L-function">L-function</a>. The <a href="/wiki/Polylogarithm" title="Polylogarithm">polylogarithm</a> is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Li</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{s}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79dcba01f9662ced9021bef451e4dfc11273940f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.892ex; height:6.843ex;" alt="{\displaystyle \operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{s}}}}"></dd></dl> which coincides with the Riemann zeta function when z = 1. The <a href="/wiki/Clausen_function" title="Clausen function">Clausen function</a> Cls(θ) can be chosen as the real or imaginary part of Lis(eiθ). The <a href="/wiki/Lerch_transcendent" title="Lerch transcendent">Lerch transcendent</a> is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (z,s,q)=\sum _{k=0}^{\infty }{\frac {z^{k}}{(k+q)^{s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>s</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (z,s,q)=\sum _{k=0}^{\infty }{\frac {z^{k}}{(k+q)^{s}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36c9cbd553de34d677b0a0e3c4ba1b859e83591c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.414ex; height:7.009ex;" alt="{\displaystyle \Phi (z,s,q)=\sum _{k=0}^{\infty }{\frac {z^{k}}{(k+q)^{s}}}}"></dd></dl> which coincides with the Riemann zeta function when z = 1 and q = 1 (the lower limit of summation in the Lerch transcendent is 0, not 1). The <a href="/wiki/Multiple_zeta_functions" class="mw-redirect" title="Multiple zeta functions">multiple zeta functions</a> are defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s_{1},s_{2},\ldots ,s_{n})=\sum _{k_{1}>k_{2}>\cdots >k_{n}>0}{k_{1}}^{-s_{1}}{k_{2}}^{-s_{2}}\cdots {k_{n}}^{-s_{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>></mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <mo>⋯</mo> <mo>></mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mn>0</mn> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>⋯</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s_{1},s_{2},\ldots ,s_{n})=\sum _{k_{1}>k_{2}>\cdots >k_{n}>0}{k_{1}}^{-s_{1}}{k_{2}}^{-s_{2}}\cdots {k_{n}}^{-s_{n}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/599ae2f60b54ef2a1bfa8579459ad841d246f0ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:52.838ex; height:5.843ex;" alt="{\displaystyle \zeta (s_{1},s_{2},\ldots ,s_{n})=\sum _{k_{1}>k_{2}>\cdots >k_{n}>0}{k_{1}}^{-s_{1}}{k_{2}}^{-s_{2}}\cdots {k_{n}}^{-s_{n}}.}"></dd></dl> One can analytically continue these functions to the n-dimensional complex space. The special values taken by these functions at positive integer arguments are called <a href="/wiki/Multiple_zeta_values" class="mw-redirect" title="Multiple zeta values">multiple zeta values</a> by number theorists and have been connected to many different branches in mathematics and physics. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=36" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7%C2%B7%C2%B7" class="mw-redirect" title="1 + 2 + 3 + 4 + ···">1 + 2 + 3 + 4 + ···</a></li> <li><a href="/wiki/Arithmetic_zeta_function" title="Arithmetic zeta function">Arithmetic zeta function</a></li> <li><a href="/wiki/Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">Generalized Riemann hypothesis</a></li> <li><a href="/wiki/Lehmer_pair" title="Lehmer pair">Lehmer pair</a></li> <li><a href="/wiki/Prime_zeta_function" title="Prime zeta function">Prime zeta function</a></li> <li><a href="/wiki/Riemann_Xi_function" class="mw-redirect" title="Riemann Xi function">Riemann Xi function</a></li> <li><a href="/wiki/Renormalization" title="Renormalization">Renormalization</a></li> <li><a href="/wiki/Riemann%E2%80%93Siegel_theta_function" title="Riemann–Siegel theta function">Riemann–Siegel theta function</a></li> <li><a href="/wiki/ZetaGrid" title="ZetaGrid">ZetaGrid</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=37" title="Edit section: References">edit</a>]</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 reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <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 class="citation web cs1"><a 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New York, Dover publications. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.212186/page/n57/mode/2up">51–55</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Functions%2C+Part+Two&rft.pages=51-55&rft.pub=New+York%2C+Dover+publications&rft.date=1947&rft.aulast=Knopp&rft.aufirst=Konrad&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.212186&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"> </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiemann1859" class="citation journal cs1">Riemann, Bernhard (1859). "<a href="/wiki/On_the_number_of_primes_less_than_a_given_magnitude" class="mw-redirect" title="On the number of primes less than a given magnitude">On the number of primes less than a given magnitude</a>". Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monatsberichte+der+K%C3%B6niglich+Preu%C3%9Fischen+Akademie+der+Wissenschaften+zu+Berlin&rft.atitle=On+the+number+of+primes+less+than+a+given+magnitude&rft.date=1859&rft.aulast=Riemann&rft.aufirst=Bernhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"> translated and reprinted in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdwards1974" class="citation book cs1"><a href="/wiki/Harold_Edwards_(mathematician)" title="Harold Edwards (mathematician)">Edwards, H. M.</a> (1974). Riemann's Zeta Function. 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Retrieved 17 April 2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=oeis.org&rft.atitle=A220335+-+OEIS&rft_id=http%3A%2F%2Foeis.org%2FA220335&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"> </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTóth2022" class="citation journal cs1"><a href="/wiki/T%C3%B3th" title="Tóth">Tóth, László</a> (2022). "Linear Combinations of Dirichlet Series Associated with the Thue-Morse Sequence". Integers. 22 (article 98). <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2211.13570">2211.13570</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Integers&rft.atitle=Linear+Combinations+of+Dirichlet+Series+Associated+with+the+Thue-Morse+Sequence&rft.volume=22&rft.issue=article+98&rft.date=2022&rft_id=info%3Aarxiv%2F2211.13570&rft.aulast=T%C3%B3th&rft.aufirst=L%C3%A1szl%C3%B3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"> </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOdlyzkoSchönhage1988" class="citation journal cs1"><a href="/wiki/Odlyzko" class="mw-redirect" title="Odlyzko">Odlyzko, A. M.</a>; <a href="/wiki/Sch%C3%B6nhage" class="mw-redirect" title="Schönhage">Schönhage, A.</a> (1988). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2000939">"Fast algorithms for multiple evaluations of the Riemann zeta function"</a>. Trans. Amer. Math. Soc. 309 (2): 797–809. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2000939">10.2307/2000939</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2000939">2000939</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0961614">0961614</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Trans.+Amer.+Math.+Soc.&rft.atitle=Fast+algorithms+for+multiple+evaluations+of+the+Riemann+zeta+function&rft.volume=309&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E797-%3C%2Fspan%3E809&rft.date=1988&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0961614%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2000939%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2000939&rft.aulast=Odlyzko&rft.aufirst=A.+M.&rft.au=Sch%C3%B6nhage%2C+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.2307%252F2000939&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988">. </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/spinchains.htm">"Work on spin-chains by A. Knauf, et. al"</a>. Empslocal.ex.ac.uk. Retrieved 4 January 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Empslocal.ex.ac.uk&rft.atitle=Work+on+spin-chains+by+A.+Knauf%2C+et.+al&rft_id=http%3A%2F%2Fempslocal.ex.ac.uk%2Fpeople%2Fstaff%2Fmrwatkin%2Fzeta%2Fspinchains.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"> </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGene_Ward_Smith" class="citation web cs1">Gene Ward Smith. <a rel="nofollow" class="external text" href="https://oeis.org/A117536">"Nearest integer to locations of increasingly large peaks of abs(zeta(0.5 + i*2*Pi/log(2)*t)) for increasing real t"</a>. The On-Line Encyclopedia of Integer Sequences. Retrieved 4 March 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Nearest+integer+to+locations+of+increasingly+large+peaks+of+abs%28zeta%280.5+%2B+i%2A2%2APi%2Flog%282%29%2At%29%29+for+increasing+real+t&rft.au=Gene+Ward+Smith&rft_id=https%3A%2F%2Foeis.org%2FA117536&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"> </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliam_A._Sethares2005" class="citation book cs1">William A. Sethares (2005). Tuning, Timbre, Spectrum, Scale (2nd ed.). Springer-Verlag London. p. 74. <q>...there are many different ways to evaluate the goodness, reasonableness, fitness, or quality of a scale...Under some measures, 12-tet is the winner, under others 19-tet appears best, 53-tet often appears among the victors...</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tuning%2C+Timbre%2C+Spectrum%2C+Scale&rft.pages=74&rft.edition=2nd&rft.pub=Springer-Verlag+London&rft.date=2005&rft.au=William+A.+Sethares&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"> </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> Most of the formulas in this section are from § 4 of J. M. Borwein et al. (2000) </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=38" title="Edit section: Sources">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 25em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApostol2010" class="citation cs1">Apostol, T.M. (2010). <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/25">"Zeta and Related Functions"</a>. In <a href="/wiki/Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.). <a href="/wiki/Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-19225-5" title="Special:BookSources/978-0-521-19225-5"><bdi>978-0-521-19225-5</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zeta+and+Related+Functions&rft.btitle=NIST+Handbook+of+Mathematical+Functions&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-0-521-19225-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2723248%23id-name%3DMR&rft.aulast=Apostol&rft.aufirst=T.M.&rft_id=http%3A%2F%2Fdlmf.nist.gov%2F25&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorweinBradleyCrandall2000" class="citation journal cs1"><a href="/wiki/Jonathan_Borwein" title="Jonathan Borwein">Borwein, Jonathan</a>; Bradley, David M.; <a href="/wiki/Richard_Crandall" title="Richard Crandall">Crandall, Richard</a> (2000). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131213171642/http://empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/borwein1.pdf">"Computational Strategies for the Riemann Zeta Function"</a> (PDF). J. Comput. Appl. Math. 121 (1–2): 247–296. <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/2000JCoAM.121..247B">2000JCoAM.121..247B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0377-0427%2800%2900336-8">10.1016/S0377-0427(00)00336-8</a>. Archived from <a rel="nofollow" class="external text" href="http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf">the original</a> (PDF) on 13 December 2013 – via University of Exeter (maths.ex.ac.uk).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Comput.+Appl.+Math.&rft.atitle=Computational+Strategies+for+the+Riemann+Zeta+Function&rft.volume=121&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E247-%3C%2Fspan%3E296&rft.date=2000&rft_id=info%3Adoi%2F10.1016%2FS0377-0427%2800%2900336-8&rft_id=info%3Abibcode%2F2000JCoAM.121..247B&rft.aulast=Borwein&rft.aufirst=Jonathan&rft.au=Bradley%2C+David+M.&rft.au=Crandall%2C+Richard&rft_id=http%3A%2F%2Fwww.maths.ex.ac.uk%2F~mwatkins%2Fzeta%2Fborwein1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCvijovićKlinowski2002" class="citation journal cs1">Cvijović, Djurdje; Klinowski, Jacek (2002). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0377-0427%2802%2900358-8">"Integral representations of the Riemann zeta function for odd-integer arguments"</a>. J. Comput. Appl. Math. 142 (2): 435–439. <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/2002JCoAM.142..435C">2002JCoAM.142..435C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0377-0427%2802%2900358-8">10.1016/S0377-0427(02)00358-8</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1906742">1906742</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Comput.+Appl.+Math.&rft.atitle=Integral+representations+of+the+Riemann+zeta+function+for+odd-integer+arguments&rft.volume=142&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E435-%3C%2Fspan%3E439&rft.date=2002&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1906742%23id-name%3DMR&rft_id=info%3Adoi%2F10.1016%2FS0377-0427%2802%2900358-8&rft_id=info%3Abibcode%2F2002JCoAM.142..435C&rft.aulast=Cvijovi%C4%87&rft.aufirst=Djurdje&rft.au=Klinowski%2C+Jacek&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252FS0377-0427%252802%252900358-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCvijovićKlinowski1997" class="citation journal cs1">Cvijović, Djurdje; Klinowski, Jacek (1997). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-97-04102-6">"Continued-fraction expansions for the Riemann zeta function and polylogarithms"</a>. Proc. Amer. Math. Soc. 125 (9): 2543–2550. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-97-04102-6">10.1090/S0002-9939-97-04102-6</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+Amer.+Math.+Soc.&rft.atitle=Continued-fraction+expansions+for+the+Riemann+zeta+function+and+polylogarithms&rft.volume=125&rft.issue=9&rft.pages=%3Cspan+class%3D%22nowrap%22%3E2543-%3C%2Fspan%3E2550&rft.date=1997&rft_id=info%3Adoi%2F10.1090%2FS0002-9939-97-04102-6&rft.aulast=Cvijovi%C4%87&rft.aufirst=Djurdje&rft.au=Klinowski%2C+Jacek&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0002-9939-97-04102-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdwards1974" class="citation book cs1"><a href="/wiki/Harold_Edwards_(mathematician)" title="Harold Edwards (mathematician)">Edwards, H.M.</a> (1974). <a rel="nofollow" class="external text" href="https://archive.org/details/riemannszetafunc00edwa_0">Riemann's Zeta Function</a>. Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-41740-9" title="Special:BookSources/0-486-41740-9"><bdi>0-486-41740-9</bdi></a> – via archive.org.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Riemann%27s+Zeta+Function&rft.pub=Academic+Press&rft.date=1974&rft.isbn=0-486-41740-9&rft.aulast=Edwards&rft.aufirst=H.M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Friemannszetafunc00edwa_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"> Has an English translation of Riemann's paper.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHadamard1896" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Jacques_Hadamard" title="Jacques Hadamard">Hadamard, Jacques</a> (1896). <a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fbsmf.545">"Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques"</a> [Regarding the distribution of the zeros of the function ζ(s) and the arithmetical consequences]. Bulletin de la Société Mathématique de France (in French). 14: 199–220. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fbsmf.545">10.24033/bsmf.545</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+de+la+Soci%C3%A9t%C3%A9+Math%C3%A9matique+de+France&rft.atitle=Sur+la+distribution+des+z%C3%A9ros+de+la+fonction+%3Cspan+class%3D%22texhtml+%22+%3E%CE%B6%28s%29%3C%2Fspan%3E+et+ses+cons%C3%A9quences+arithm%C3%A9tiques&rft.volume=14&rft.pages=%3Cspan+class%3D%22nowrap%22%3E199-%3C%2Fspan%3E220&rft.date=1896&rft_id=info%3Adoi%2F10.24033%2Fbsmf.545&rft.aulast=Hadamard&rft.aufirst=Jacques&rft_id=https%3A%2F%2Fdoi.org%2F10.24033%252Fbsmf.545&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardy1949" class="citation book cs1"><a href="/wiki/G._H._Hardy" title="G. H. Hardy">Hardy, G.H.</a> (1949). Divergent Series. Oxford, UK: Clarendon Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Divergent+Series&rft.place=Oxford%2C+UK&rft.pub=Clarendon+Press&rft.date=1949&rft.aulast=Hardy&rft.aufirst=G.H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHasse1930" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Helmut_Hasse" title="Helmut Hasse">Hasse, Helmut</a> (1930). "Ein Summierungsverfahren für die Riemannsche ζ-Reihe" [A summation method for the Riemann ζ series]. Math. Z. (in German). 32: 458–464. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01194645">10.1007/BF01194645</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1545177">1545177</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120392534">120392534</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math.+Z.&rft.atitle=Ein+Summierungsverfahren+f%C3%BCr+die+Riemannsche+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CE%B6%3C%2Fspan%3E-Reihe&rft.volume=32&rft.pages=%3Cspan+class%3D%22nowrap%22%3E458-%3C%2Fspan%3E464&rft.date=1930&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1545177%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120392534%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01194645&rft.aulast=Hasse&rft.aufirst=Helmut&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"> (Globally convergent series expression.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIvic1985" class="citation book cs1"><a href="/wiki/Aleksandar_Ivi%C4%87" title="Aleksandar Ivić">Ivic, Aleksandar</a> (1985). The Riemann Zeta Function. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-80634-X" title="Special:BookSources/0-471-80634-X"><bdi>0-471-80634-X</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+Zeta+Function&rft.pub=John+Wiley+%26+Sons&rft.date=1985&rft.isbn=0-471-80634-X&rft.aulast=Ivic&rft.aufirst=Aleksandar&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMotohashi1997" class="citation book cs1">Motohashi, Y. (1997). Spectral Theory of the Riemann Zeta-Function. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0521445205" title="Special:BookSources/0521445205"><bdi>0521445205</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spectral+Theory+of+the+Riemann+Zeta-Function&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=0521445205&rft.aulast=Motohashi&rft.aufirst=Y.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaratsubaVoronin1992" class="citation book cs1"><a href="/wiki/A._A._Karatsuba" class="mw-redirect" title="A. A. Karatsuba">Karatsuba, A.A.</a>; Voronin, S.M. (1992). The Riemann Zeta-Function. Berlin, DE: W. de Gruyter.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Riemann+Zeta-Function&rft.place=Berlin%2C+DE&rft.pub=W.+de+Gruyter&rft.date=1992&rft.aulast=Karatsuba&rft.aufirst=A.A.&rft.au=Voronin%2C+S.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMezőDil2010" class="citation journal cs1">Mező, István; Dil, Ayhan (2010). "Hyperharmonic series involving Hurwitz zeta function". Journal of Number Theory. 130 (2): 360–369. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jnt.2009.08.005">10.1016/j.jnt.2009.08.005</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2437%2F90539">2437/90539</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2564902">2564902</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122707401">122707401</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Number+Theory&rft.atitle=Hyperharmonic+series+involving+Hurwitz+zeta+function&rft.volume=130&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E360-%3C%2Fspan%3E369&rft.date=2010&rft_id=info%3Ahdl%2F2437%2F90539&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2564902%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122707401%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.jnt.2009.08.005&rft.aulast=Mez%C5%91&rft.aufirst=Istv%C3%A1n&rft.au=Dil%2C+Ayhan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMontgomeryVaughan2007" class="citation book cs1"><a href="/wiki/Hugh_Montgomery_(mathematician)" class="mw-redirect" title="Hugh Montgomery (mathematician)">Montgomery, Hugh L.</a>; <a href="/wiki/Robert_Charles_Vaughan_(mathematician)" class="mw-redirect" title="Robert Charles Vaughan (mathematician)">Vaughan, Robert C.</a> (2007). Multiplicative Number Theory. I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge University Press. Chapter 10. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-84903-6" title="Special:BookSources/978-0-521-84903-6"><bdi>978-0-521-84903-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multiplicative+Number+Theory.+I.+Classical+theory&rft.series=Cambridge+tracts+in+advanced+mathematics&rft.pages=Chapter-10&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-84903-6&rft.aulast=Montgomery&rft.aufirst=Hugh+L.&rft.au=Vaughan%2C+Robert+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNewman1998" class="citation book cs1"><a href="/wiki/Donald_J._Newman" title="Donald J. Newman">Newman, Donald J.</a> (1998). Analytic Number Theory. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. Vol. 177. Springer-Verlag. Ch. 6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98308-2" title="Special:BookSources/0-387-98308-2"><bdi>0-387-98308-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytic+Number+Theory&rft.series=Graduate+Texts+in+Mathematics&rft.pages=Ch.+6&rft.pub=Springer-Verlag&rft.date=1998&rft.isbn=0-387-98308-2&rft.aulast=Newman&rft.aufirst=Donald+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRaoh1996" class="citation journal cs1 cs1-prop-long-vol">Raoh, Guo (1996). "The distribution of the logarithmic derivative of the Riemann zeta function". Proceedings of the London Mathematical Society. S3–72: 1–27. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs3-72.1.1">10.1112/plms/s3-72.1.1</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+London+Mathematical+Society&rft.atitle=The+distribution+of+the+logarithmic+derivative+of+the+Riemann+zeta+function&rft.volume=S3%E2%80%9372&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E27&rft.date=1996&rft_id=info%3Adoi%2F10.1112%2Fplms%2Fs3-72.1.1&rft.aulast=Raoh&rft.aufirst=Guo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiemann1859" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann, Bernhard</a> (1859). <a rel="nofollow" class="external text" href="http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/">"Über die Anzahl der Primzahlen unter einer gegebenen Grösse"</a>. Monatsberichte der Berliner Akademie (in German and English) – via <a href="/wiki/Trinity_College,_Dublin" class="mw-redirect" title="Trinity College, Dublin">Trinity College, Dublin</a> (maths.tcd.ie).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monatsberichte+der+Berliner+Akademie&rft.atitle=%C3%9Cber+die+Anzahl+der+Primzahlen+unter+einer+gegebenen+Gr%C3%B6sse&rft.date=1859&rft.aulast=Riemann&rft.aufirst=Bernhard&rft_id=http%3A%2F%2Fwww.maths.tcd.ie%2Fpub%2FHistMath%2FPeople%2FRiemann%2FZeta%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"> Also available in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiemann1953" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann, Bernhard</a> (1953) [1892]. Gesammelte Werke [Collected Works] (in German) (reprint ed.). New York, NY / Leipzig, DE: Dover (1953) / Teubner (1892).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gesammelte+Werke&rft.place=New+York%2C+NY+%2F+Leipzig%2C+DE&rft.edition=reprint&rft.pub=Dover+%281953%29+%2F+Teubner+%281892%29&rft.date=1953&rft.aulast=Riemann&rft.aufirst=Bernhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSondow1994" class="citation journal cs1">Sondow, Jonathan (1994). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/proc/1994-120-02/S0002-9939-1994-1172954-7/S0002-9939-1994-1172954-7.pdf">"Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series"</a> (PDF). <a href="/wiki/Proceedings_of_the_American_Mathematical_Society" title="Proceedings of the American Mathematical Society">Proceedings of the American Mathematical Society</a>. 120 (2): 421–424. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-1994-1172954-7">10.1090/S0002-9939-1994-1172954-7</a> – via <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a> (ams.org).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=Analytic+continuation+of+Riemann%27s+zeta+function+and+values+at+negative+integers+via+Euler%27s+transformation+of+series&rft.volume=120&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E421-%3C%2Fspan%3E424&rft.date=1994&rft_id=info%3Adoi%2F10.1090%2FS0002-9939-1994-1172954-7&rft.aulast=Sondow&rft.aufirst=Jonathan&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fproc%2F1994-120-02%2FS0002-9939-1994-1172954-7%2FS0002-9939-1994-1172954-7.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTitchmarsh1986" class="citation book cs1"><a href="/wiki/Edward_Charles_Titchmarsh" title="Edward Charles Titchmarsh">Titchmarsh, E.C.</a> (1986). Heath-Brown (ed.). The Theory of the Riemann Zeta Function (2nd rev. ed.). Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+the+Riemann+Zeta+Function&rft.edition=2nd+rev.&rft.pub=Oxford+University+Press&rft.date=1986&rft.aulast=Titchmarsh&rft.aufirst=E.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhittakerWatson1927" class="citation book cs1"><a href="/wiki/E._T._Whittaker" title="E. T. Whittaker">Whittaker, E.T.</a>; <a href="/wiki/G._N._Watson" title="G. N. Watson">Watson, G.N.</a> (1927). A Course in Modern Analysis (4th ed.). Cambridge University Press. Chapter 13.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Modern+Analysis&rft.pages=Chapter-13&rft.edition=4th&rft.pub=Cambridge+University+Press&rft.date=1927&rft.aulast=Whittaker&rft.aufirst=E.T.&rft.au=Watson%2C+G.N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZhao1999" class="citation journal cs1">Zhao, Jianqiang (1999). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-99-05398-8">"Analytic continuation of multiple zeta functions"</a>. <a href="/wiki/Proceedings_of_the_American_Mathematical_Society" title="Proceedings of the American Mathematical Society">Proceedings of the American Mathematical Society</a>. 128 (5): 1275–1283. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-99-05398-8">10.1090/S0002-9939-99-05398-8</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1670846">1670846</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=Analytic+continuation+of+multiple+zeta+functions&rft.volume=128&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1275-%3C%2Fspan%3E1283&rft.date=1999&rft_id=info%3Adoi%2F10.1090%2FS0002-9939-99-05398-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1670846%23id-name%3DMR&rft.aulast=Zhao&rft.aufirst=Jianqiang&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0002-9939-99-05398-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Riemann_zeta_function&action=edit&section=39" title="Edit section: External links">edit</a>]</div> <ul><li><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Riemann_zeta_function" class="extiw" title="commons:Category:Riemann zeta function">Riemann zeta function</a> at Wikimedia Commons</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Zeta-function">"Zeta-function"</a>. <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>. <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>. 2001 [1994].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zeta-function&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DZeta-function&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/RiemannZetaFunction.html">Riemann Zeta Function, in Wolfram Mathworld</a> — an explanation with a more mathematical approach</li> <li><a rel="nofollow" class="external text" href="http://dtc.umn.edu/~odlyzko/zeta_tables">Tables of selected zeros</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090517003700/http://dtc.umn.edu/~odlyzko/zeta_tables/">Archived</a> 17 May 2009 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080721030342/http://seedmagazine.com/news/2006/03/prime_numbers_get_hitched.php">Prime Numbers Get Hitched</a> A general, non-technical description of the significance of the zeta function in relation to prime numbers.</li> <li><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0309433v1">X-Ray of the Zeta Function</a> Visually oriented investigation of where zeta is real or purely imaginary.</li> <li><a rel="nofollow" class="external text" href="http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/">Formulas and identities for the Riemann Zeta function</a> functions.wolfram.com</li> <li><a rel="nofollow" class="external text" href="http://www.math.sfu.ca/~cbm/aands/page_807.htm">Riemann Zeta Function and Other Sums of Reciprocal Powers</a>, section 23.2 of <a href="/wiki/Abramowitz_and_Stegun" title="Abramowitz and Stegun">Abramowitz and Stegun</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrenkel" class="citation web cs1"><a href="/wiki/Edward_Frenkel" title="Edward Frenkel">Frenkel, Edward</a>. <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=d6c6uIyieoo">"Million Dollar Math Problem"</a> (video). <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211211/d6c6uIyieoo">Archived</a> from the original on 11 December 2021. Retrieved 11 March 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Million+Dollar+Math+Problem&rft.pub=Brady+Haran&rft.aulast=Frenkel&rft.aufirst=Edward&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3Dd6c6uIyieoo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiemann+zeta+function" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://www.mathematik.uni-stuttgart.de/~riedelmo/papers/rfeq.pdf">Mellin transform and the functional equation of the Riemann Zeta function</a>—Computational examples of Mellin transform methods involving the Riemann Zeta Function</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=sD0NjbwqlYw">Visualizing the Riemann zeta function and analytic continuation</a> a video from <a href="/wiki/3Blue1Brown" title="3Blue1Brown">3Blue1Brown</a></li></ul> <div class="navbox-styles"><style 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theory">number theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Analytic examples</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Riemann zeta function</a></li> <li><a href="/wiki/Dirichlet_L-function" title="Dirichlet L-function">Dirichlet L-functions</a></li> <li><a href="/wiki/L-function_with_Gr%C3%B6ssencharakter" class="mw-redirect" title="L-function with Grössencharakter">L-functions of Hecke characters</a></li> <li><a href="/wiki/Automorphic_L-function" title="Automorphic L-function">Automorphic L-functions</a></li> <li><a href="/wiki/Selberg_class" title="Selberg class">Selberg class</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebraic examples</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dedekind_zeta_function" title="Dedekind zeta function">Dedekind zeta functions</a></li> <li><a href="/wiki/Artin_L-function" title="Artin L-function">Artin L-functions</a></li> <li><a href="/wiki/Hasse%E2%80%93Weil_zeta_function" title="Hasse–Weil zeta function">Hasse–Weil L-functions</a></li> <li><a href="/wiki/Motivic_L-function" title="Motivic L-function">Motivic L-functions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Class_number_formula" title="Class number formula">Analytic class number formula</a></li> <li><a href="/wiki/Riemann%E2%80%93von_Mangoldt_formula" title="Riemann–von Mangoldt formula">Riemann–von Mangoldt formula</a></li> <li><a href="/wiki/Weil_conjectures" title="Weil conjectures">Weil conjectures</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Analytic conjectures</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li> <li><a href="/wiki/Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">Generalized Riemann hypothesis</a></li> <li><a href="/wiki/Lindel%C3%B6f_hypothesis" title="Lindelöf hypothesis">Lindelöf hypothesis</a></li> <li><a href="/wiki/Ramanujan%E2%80%93Petersson_conjecture" title="Ramanujan–Petersson conjecture">Ramanujan–Petersson conjecture</a></li> <li><a href="/wiki/Artin_conjecture_(L-functions)" class="mw-redirect" title="Artin conjecture (L-functions)">Artin conjecture</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebraic conjectures</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Birch_and_Swinnerton-Dyer_conjecture" title="Birch and Swinnerton-Dyer conjecture">Birch and Swinnerton-Dyer conjecture</a></li> <li><a href="/wiki/Special_values_of_L-functions" title="Special values of L-functions">Deligne's conjecture</a></li> <li><a href="/wiki/Beilinson_conjectures" class="mw-redirect" title="Beilinson conjectures">Beilinson conjectures</a></li> <li><a href="/wiki/Bloch%E2%80%93Kato_conjecture_(L-functions)" class="mw-redirect" title="Bloch–Kato conjecture (L-functions)">Bloch–Kato conjecture</a></li> <li><a href="/wiki/Langlands_program" title="Langlands program">Langlands conjecture</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/P-adic_L-function" title="P-adic L-function">p-adic L-functions</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Main_conjecture_of_Iwasawa_theory" title="Main conjecture of Iwasawa theory">Main conjecture of Iwasawa theory</a></li> <li><a href="/wiki/Selmer_group" title="Selmer group">Selmer group</a></li> <li><a href="/wiki/Euler_system" title="Euler system">Euler system</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Sequences_and_series" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Series_(mathematics)" title="Template:Series (mathematics)"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Series_(mathematics)" title="Template talk:Series (mathematics)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Series_(mathematics)" title="Special:EditPage/Template:Series (mathematics)"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Sequences_and_series" style="font-size:114%;margin:0 4em"><a href="/wiki/Sequence" title="Sequence">Sequences</a> and <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integer_sequence" title="Integer sequence">Integer sequences</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Basic</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic_progression" title="Arithmetic progression">Arithmetic progression</a></li> <li><a href="/wiki/Geometric_progression" title="Geometric progression">Geometric progression</a></li> <li><a href="/wiki/Harmonic_progression_(mathematics)" title="Harmonic progression (mathematics)">Harmonic progression</a></li> <li><a href="/wiki/Square_number" title="Square number">Square number</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic number</a></li> <li><a href="/wiki/Factorial" title="Factorial">Factorial</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Powers of two</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Powers of three</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Powers of 10</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Advanced (<a href="/wiki/List_of_OEIS_sequences" class="mw-redirect" title="List of OEIS sequences">list</a>)</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Complete_sequence" title="Complete sequence">Complete sequence</a></li> <li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a></li> <li><a href="/wiki/Figurate_number" title="Figurate number">Figurate number</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal number</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal number</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas number</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell number</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal number</a></li> <li><a href="/wiki/Polygonal_number" title="Polygonal number">Polygonal number</a></li> <li><a href="/wiki/Triangular_number" title="Triangular number">Triangular number</a> <ul><li><a href="/wiki/Triangular_array" title="Triangular array">array</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence"><img alt="Fibonacci spiral with square sizes up to 34." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/80px-Fibonacci_spiral_34.svg.png" decoding="async" width="80" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/120px-Fibonacci_spiral_34.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/160px-Fibonacci_spiral_34.svg.png 2x" data-file-width="915" data-file-height="579" /></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of sequences</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></li> <li><a href="/wiki/Monotonic_function" title="Monotonic function">Monotonic function</a></li> <li><a href="/wiki/Periodic_sequence" title="Periodic sequence">Periodic sequence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Series</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Convergent_series" title="Convergent series">Convergent</a></li> <li><a href="/wiki/Divergent_series" title="Divergent series">Divergent</a></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergence</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_convergence" title="Absolute convergence">Absolute</a></li> <li><a href="/wiki/Conditional_convergence" title="Conditional convergence">Conditional</a></li> <li><a href="/wiki/Uniform_convergence" title="Uniform convergence">Uniform</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Explicit series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergent</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/1/2_%E2%88%92_1/4_%2B_1/8_%E2%88%92_1/16_%2B_%E2%8B%AF" title="1/2 − 1/4 + 1/8 − 1/16 + ⋯">1/2 − 1/4 + 1/8 − 1/16 + ⋯</a></li> <li><a href="/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_%E2%8B%AF" title="1/2 + 1/4 + 1/8 + 1/16 + ⋯">1/2 + 1/4 + 1/8 + 1/16 + ⋯</a></li> <li><a href="/wiki/1/4_%2B_1/16_%2B_1/64_%2B_1/256_%2B_%E2%8B%AF" title="1/4 + 1/16 + 1/64 + 1/256 + ⋯">1/4 + 1/16 + 1/64 + 1/256 + ⋯</a></li> <li><a class="mw-selflink selflink">1 + 1/2s + 1/3s + ... (Riemann zeta function)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Divergent</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/1_%2B_1_%2B_1_%2B_1_%2B_%E2%8B%AF" title="1 + 1 + 1 + 1 + ⋯">1 + 1 + 1 + 1 + ⋯</a></li> <li><a href="/wiki/Grandi%27s_series" title="Grandi's series">1 − 1 + 1 − 1 + ⋯ (Grandi's series)</a></li> <li><a href="/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯">1 + 2 + 3 + 4 + ⋯</a></li> <li><a href="/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%E2%8B%AF" title="1 − 2 + 3 − 4 + ⋯">1 − 2 + 3 − 4 + ⋯</a></li> <li><a href="/wiki/1_%2B_2_%2B_4_%2B_8_%2B_%E2%8B%AF" title="1 + 2 + 4 + 8 + ⋯">1 + 2 + 4 + 8 + ⋯</a></li> <li><a href="/wiki/1_%E2%88%92_2_%2B_4_%E2%88%92_8_%2B_%E2%8B%AF" title="1 − 2 + 4 − 8 + ⋯">1 − 2 + 4 − 8 + ⋯</a></li> <li><a href="/wiki/Infinite_arithmetic_series" class="mw-redirect" title="Infinite arithmetic series">Infinite arithmetic series</a></li> <li><a href="/wiki/1_%E2%88%92_1_%2B_2_%E2%88%92_6_%2B_24_%E2%88%92_120_%2B_..." class="mw-redirect" title="1 − 1 + 2 − 6 + 24 − 120 + ...">1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)</a></li> <li><a href="/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes" title="Divergence of the sum of the reciprocals of the primes">1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Kinds of series</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a></li> <li><a href="/wiki/Power_series" title="Power series">Power series</a></li> <li><a href="/wiki/Formal_power_series" title="Formal power series">Formal power series</a></li> <li><a href="/wiki/Laurent_series" title="Laurent series">Laurent series</a></li> <li><a href="/wiki/Puiseux_series" title="Puiseux series">Puiseux series</a></li> <li><a href="/wiki/Dirichlet_series" title="Dirichlet series">Dirichlet series</a></li> <li><a href="/wiki/Trigonometric_series" title="Trigonometric series">Trigonometric series</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a></li> <li><a href="/wiki/Generating_series" class="mw-redirect" title="Generating series">Generating series</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hypergeometric_function" title="Hypergeometric function">Hypergeometric series</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Generalized_hypergeometric_series" class="mw-redirect" title="Generalized hypergeometric series">Generalized hypergeometric series</a></li> <li><a href="/wiki/Hypergeometric_function_of_a_matrix_argument" title="Hypergeometric function of a matrix argument">Hypergeometric function of a matrix argument</a></li> <li><a href="/wiki/Lauricella_hypergeometric_series" title="Lauricella hypergeometric series">Lauricella hypergeometric series</a></li> <li><a href="/wiki/Modular_hypergeometric_series" class="mw-redirect" title="Modular hypergeometric series">Modular hypergeometric series</a></li> <li><a href="/wiki/Riemann%27s_differential_equation" title="Riemann's differential equation">Riemann's differential equation</a></li> <li><a href="/wiki/Theta_hypergeometric_series" class="mw-redirect" title="Theta hypergeometric series">Theta hypergeometric series</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Mathematical_series" title="Category:Mathematical series">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Bernhard_Riemann" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Bernhard_Riemann" title="Template:Bernhard Riemann"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Bernhard_Riemann" title="Template talk:Bernhard Riemann"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Bernhard_Riemann" title="Special:EditPage/Template:Bernhard Riemann"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Bernhard_Riemann" style="font-size:114%;margin:0 4em"><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a></li> <li><a href="/wiki/Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">Generalized Riemann hypothesis</a></li> <li><a href="/wiki/Grand_Riemann_hypothesis" title="Grand Riemann hypothesis">Grand Riemann hypothesis</a></li> <li><a href="/wiki/Grothendieck%E2%80%93Riemann%E2%80%93Roch_theorem" title="Grothendieck–Riemann–Roch theorem">Grothendieck–Hirzebruch–Riemann–Roch theorem</a></li> <li><a href="/wiki/Hirzebruch%E2%80%93Riemann%E2%80%93Roch_theorem" title="Hirzebruch–Riemann–Roch theorem">Hirzebruch–Riemann–Roch theorem</a></li> <li><a href="/wiki/Local_zeta_function" title="Local zeta function">Local zeta function</a></li> <li><a href="/wiki/Measurable_Riemann_mapping_theorem" title="Measurable Riemann mapping theorem">Measurable Riemann mapping theorem</a></li> <li><a href="/wiki/Riemann_(crater)" title="Riemann (crater)">Riemann (crater)</a></li> <li><a href="/wiki/Riemann_Xi_function" class="mw-redirect" title="Riemann Xi function">Riemann Xi function</a></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Riemann_invariant" title="Riemann invariant">Riemann invariant</a></li> <li><a href="/wiki/Riemann_mapping_theorem" title="Riemann mapping theorem">Riemann mapping theorem</a></li> <li><a href="/wiki/Riemann_form" title="Riemann form">Riemann form</a></li> <li><a href="/wiki/Riemann_problem" title="Riemann problem">Riemann problem</a></li> <li><a href="/wiki/Riemann_series_theorem" title="Riemann series theorem">Riemann series theorem</a></li> <li><a href="/wiki/Riemann_solver" title="Riemann solver">Riemann solver</a></li> <li><a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a></li> <li><a href="/wiki/Riemann_sum" title="Riemann sum">Riemann sum</a></li> <li><a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surface</a></li> <li><a class="mw-selflink selflink">Riemann zeta function</a></li> <li><a href="/wiki/Riemann%27s_differential_equation" title="Riemann's differential equation">Riemann's differential equation</a></li> <li><a href="/wiki/Riemann%27s_minimal_surface" title="Riemann's minimal surface">Riemann's minimal surface</a></li> <li><a href="/wiki/Riemannian_circle" class="mw-redirect" title="Riemannian circle">Riemannian circle</a></li> <li><a href="/wiki/Riemannian_connection_on_a_surface" title="Riemannian connection on a surface">Riemannian connection on a surface</a></li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian 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title="Riemann–Siegel theta function">Riemann–Siegel theta function</a></li> <li><a href="/wiki/Riemann%E2%80%93Silberstein_vector" title="Riemann–Silberstein vector">Riemann–Silberstein vector</a></li> <li><a href="/wiki/Riemann%E2%80%93Stieltjes_integral" title="Riemann–Stieltjes integral">Riemann–Stieltjes integral</a></li> <li><a href="/wiki/Riemann%E2%80%93von_Mangoldt_formula" title="Riemann–von Mangoldt formula">Riemann–von Mangoldt formula</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" 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Riemann zeta function - Wikipedia