Euler's totient function

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class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Computing Euler's totient function subsection </button> <ul id="toc-Computing_Euler's_totient_function-sublist" class="vector-toc-list"> <li id="toc-Euler's_product_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Euler's_product_formula"> <div class="vector-toc-text"> 2.1 Euler's product formula </div> </a> <ul id="toc-Euler's_product_formula-sublist" class="vector-toc-list"> <li id="toc-Phi_is_a_multiplicative_function" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Phi_is_a_multiplicative_function"> <div class="vector-toc-text"> 2.1.1 Phi is a multiplicative function </div> </a> <ul id="toc-Phi_is_a_multiplicative_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Value_of_phi_for_a_prime_power_argument" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Value_of_phi_for_a_prime_power_argument"> <div class="vector-toc-text"> 2.1.2 Value of phi for a prime power argument </div> </a> <ul id="toc-Value_of_phi_for_a_prime_power_argument-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proof_of_Euler's_product_formula" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Proof_of_Euler's_product_formula"> <div class="vector-toc-text"> 2.1.3 Proof of Euler's product formula </div> </a> <ul id="toc-Proof_of_Euler's_product_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> 2.1.4 Example </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Fourier_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fourier_transform"> <div class="vector-toc-text"> 2.2 Fourier transform </div> </a> <ul id="toc-Fourier_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Divisor_sum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Divisor_sum"> <div class="vector-toc-text"> 2.3 Divisor sum </div> </a> <ul id="toc-Divisor_sum-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Some_values" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Some_values"> <div class="vector-toc-text"> 3 Some values </div> </a> <ul id="toc-Some_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euler's_theorem" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Euler's_theorem"> <div class="vector-toc-text"> 4 Euler's theorem </div> </a> <ul id="toc-Euler's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_formulae" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_formulae"> <div class="vector-toc-text"> 5 Other formulae </div> </a> <button aria-controls="toc-Other_formulae-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Other formulae subsection </button> <ul id="toc-Other_formulae-sublist" class="vector-toc-list"> <li id="toc-Menon's_identity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Menon's_identity"> <div class="vector-toc-text"> 5.1 Menon's identity </div> </a> <ul id="toc-Menon's_identity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Divisibility_by_any_fixed_positive_integer" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Divisibility_by_any_fixed_positive_integer"> <div class="vector-toc-text"> 5.2 Divisibility by any fixed positive integer </div> </a> <ul id="toc-Divisibility_by_any_fixed_positive_integer-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generating_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generating_functions"> <div class="vector-toc-text"> 6 Generating functions </div> </a> <ul id="toc-Generating_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Growth_rate" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Growth_rate"> <div class="vector-toc-text"> 7 Growth rate </div> </a> <ul id="toc-Growth_rate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ratio_of_consecutive_values" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ratio_of_consecutive_values"> <div class="vector-toc-text"> 8 Ratio of consecutive values </div> </a> <ul id="toc-Ratio_of_consecutive_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Totient_numbers" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Totient_numbers"> <div class="vector-toc-text"> 9 Totient numbers </div> </a> <button aria-controls="toc-Totient_numbers-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Totient numbers subsection </button> <ul id="toc-Totient_numbers-sublist" class="vector-toc-list"> <li id="toc-Ford's_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ford's_theorem"> <div class="vector-toc-text"> 9.1 Ford's theorem </div> </a> <ul id="toc-Ford's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Perfect_totient_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Perfect_totient_numbers"> <div class="vector-toc-text"> 9.2 Perfect totient numbers </div> </a> <ul id="toc-Perfect_totient_numbers-sublist" class="vector-toc-list"> </ul> </li> </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"> 10 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-Cyclotomy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cyclotomy"> <div class="vector-toc-text"> 10.1 Cyclotomy </div> </a> <ul id="toc-Cyclotomy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prime_number_theorem_for_arithmetic_progressions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prime_number_theorem_for_arithmetic_progressions"> <div class="vector-toc-text"> 10.2 Prime number theorem for arithmetic progressions </div> </a> <ul id="toc-Prime_number_theorem_for_arithmetic_progressions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_RSA_cryptosystem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_RSA_cryptosystem"> <div class="vector-toc-text"> 10.3 The RSA cryptosystem </div> </a> <ul id="toc-The_RSA_cryptosystem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Unsolved_problems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Unsolved_problems"> <div class="vector-toc-text"> 11 Unsolved problems </div> </a> <button aria-controls="toc-Unsolved_problems-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Unsolved problems subsection </button> <ul id="toc-Unsolved_problems-sublist" class="vector-toc-list"> <li id="toc-Lehmer's_conjecture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lehmer's_conjecture"> <div class="vector-toc-text"> 11.1 Lehmer's conjecture </div> </a> <ul id="toc-Lehmer's_conjecture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Carmichael's_conjecture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Carmichael's_conjecture"> <div class="vector-toc-text"> 11.2 Carmichael's conjecture </div> </a> <ul id="toc-Carmichael's_conjecture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Riemann_hypothesis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Riemann_hypothesis"> <div class="vector-toc-text"> 11.3 Riemann hypothesis </div> </a> <ul id="toc-Riemann_hypothesis-sublist" class="vector-toc-list"> </ul> </li> </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"> 12 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 13 Notes </div> </a> <ul id="toc-Notes-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"> 14 References </div> </a> <ul id="toc-References-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"> 15 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">Euler's totient 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 41 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-41" aria-hidden="true" > 41 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/%D9%85%D8%A4%D8%B4%D8%B1_%D8%A3%D9%88%D9%8A%D9%84%D8%B1" title="مؤشر أويلر – Arabic" lang="ar" hreflang="ar" data-title="مؤشر أويلر" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AF%E0%A6%BC%E0%A6%B2%E0%A6%BE%E0%A6%B0_%E0%A6%9F%E0%A7%8B%E0%A6%B6%E0%A7%87%E0%A6%A8%E0%A7%8D%E0%A6%9F_%E0%A6%AB%E0%A6%BE%E0%A6%82%E0%A6%B6%E0%A6%A8" title="অয়লার টোশেন্ট ফাংশন – Bangla" lang="bn" hreflang="bn" data-title="অয়লার টোশেন্ট ফাংশন" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%9E%D0%B9%D0%BB%D0%B5%D1%80" title="Функция на Ойлер – Bulgarian" lang="bg" hreflang="bg" data-title="Функция на Ойлер" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_%CF%86_d%27Euler" title="Funció φ d'Euler – Catalan" lang="ca" hreflang="ca" data-title="Funció φ d'Euler" 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/Eulerova_funkce" title="Eulerova funkce – Czech" lang="cs" hreflang="cs" data-title="Eulerova funkce" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ffwythiant_%CF%86_Euler" title="Ffwythiant φ Euler – Welsh" lang="cy" hreflang="cy" data-title="Ffwythiant φ Euler" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Eulers_totientfunktion" title="Eulers totientfunktion – Danish" lang="da" hreflang="da" data-title="Eulers totientfunktion" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Eulersche_Phi-Funktion" title="Eulersche Phi-Funktion – German" lang="de" hreflang="de" data-title="Eulersche Phi-Funktion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</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%8C%CE%B9%CE%BB%CE%B5%CF%81" title="Συνάρτηση Όιλερ – Greek" lang="el" hreflang="el" data-title="Συνάρτηση Όιλερ" 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_%CF%86_de_Euler" title="Función φ de Euler – Spanish" lang="es" hreflang="es" data-title="Función φ de Euler" 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/Funkcio_%CF%86" title="Funkcio φ – Esperanto" lang="eo" hreflang="eo" data-title="Funkcio φ" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Eulerren_%CF%86_funtzioa" title="Eulerren φ funtzioa – Basque" lang="eu" hreflang="eu" data-title="Eulerren φ funtzioa" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</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_%D9%81%DB%8C_%D8%A7%D9%88%DB%8C%D9%84%D8%B1" title="تابع فی اویلر – Persian" lang="fa" hreflang="fa" data-title="تابع فی اویلر" 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/Indicatrice_d%27Euler" title="Indicatrice d'Euler – French" lang="fr" hreflang="fr" data-title="Indicatrice d'Euler" 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_totiente_de_Euler" title="Función totiente de Euler – Galician" lang="gl" hreflang="gl" data-title="Función totiente de Euler" 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/%EC%98%A4%EC%9D%BC%EB%9F%AC_%ED%94%BC_%ED%95%A8%EC%88%98" title="오일러 피 함수 – Korean" lang="ko" hreflang="ko" data-title="오일러 피 함수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Eulerova_funkcija" title="Eulerova funkcija – Croatian" lang="hr" hreflang="hr" data-title="Eulerova 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_phi_Euler" title="Fungsi phi Euler – Indonesian" lang="id" hreflang="id" data-title="Fungsi phi Euler" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_%CF%86_di_Eulero" title="Funzione φ di Eulero – Italian" lang="it" hreflang="it" data-title="Funzione φ di Eulero" 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%90%D7%95%D7%99%D7%9C%D7%A8" title="פונקציית אוילר – Hebrew" lang="he" hreflang="he" data-title="פונקציית אוילר" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F%D1%81%D1%8B" title="Эйлер функциясы – Kazakh" lang="kk" hreflang="kk" data-title="Эйлер функциясы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Fonksyon_phi_Euler" title="Fonksyon phi Euler – Haitian Creole" lang="ht" hreflang="ht" data-title="Fonksyon phi Euler" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Euler-f%C3%BCggv%C3%A9ny" title="Euler-függvény – Hungarian" lang="hu" hreflang="hu" data-title="Euler-függvény" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%93%E0%B4%AF%E0%B5%8D%E0%B4%B2%E0%B4%B1%E0%B5%81%E0%B4%9F%E0%B5%86_%E0%B4%9F%E0%B5%8B%E0%B4%B7%E0%B5%8D%E0%B4%AF%E0%B4%A8%E0%B5%8D%E0%B4%B1%E0%B5%8D_%E0%B4%AB%E0%B4%B2%E0%B4%A8%E0%B4%82" title="ഓയ്ലറുടെ ടോഷ്യന്റ് ഫലനം – Malayalam" lang="ml" hreflang="ml" data-title="ഓയ്ലറുടെ ടോഷ്യന്റ് ഫലനം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Indicator_(getaltheorie)" title="Indicator (getaltheorie) – Dutch" lang="nl" hreflang="nl" data-title="Indicator (getaltheorie)" 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%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E3%81%AE%CF%86%E9%96%A2%E6%95%B0" title="オイラーのφ関数 – Japanese" lang="ja" hreflang="ja" data-title="オイラーのφ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/Eulers_totientfunksjon" title="Eulers totientfunksjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Eulers totientfunksjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja_%CF%86" title="Funkcja φ – Polish" lang="pl" hreflang="pl" data-title="Funkcja φ" 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_totiente_de_Euler" title="Função totiente de Euler – Portuguese" 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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">"φ(n)" redirects here. For other uses, see <a href="/wiki/Phi" title="Phi">Phi</a>. Not to be confused with <a href="/wiki/Euler_function" title="Euler function">Euler function</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:EulerPhi.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/EulerPhi.svg/220px-EulerPhi.svg.png" decoding="async" width="220" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/EulerPhi.svg/330px-EulerPhi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/EulerPhi.svg/440px-EulerPhi.svg.png 2x" data-file-width="731" data-file-height="551" /></a><figcaption>The first thousand values of φ(n). The points on the top line represent φ(p) when p is a prime number, which is p − 1.<a href="#cite_note-1">[1]</a></figcaption></figure> In <a href="/wiki/Number_theory" title="Number theory">number theory</a>, Euler's totient function counts the positive integers up to a given integer n that are <a href="/wiki/Relatively_prime" class="mw-redirect" title="Relatively prime">relatively prime</a> to n. It is written using the Greek letter <a href="/wiki/Phi" title="Phi">phi</a> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f067864064667dd5f8b2508b9cbf983d89788629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle \varphi (n)}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb3dbe542ca7d51c4f32e32cefb8572edb26ab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.589ex; height:2.843ex;" alt="{\displaystyle \phi (n)}">, and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> gcd(n, k) is equal to 1.<a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a> The integers k of this form are sometimes referred to as <a href="/wiki/Totative" title="Totative">totatives</a> of n. For example, the totatives of n = 9 are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. Therefore, φ(9) = 6. As another example, φ(1) = 1 since for n = 1 the only integer in the range from 1 to n is 1 itself, and gcd(1, 1) = 1. Euler's totient function is a <a href="/wiki/Multiplicative_function" title="Multiplicative function">multiplicative function</a>, meaning that if two numbers m and n are relatively prime, then φ(mn) = φ(m)φ(n).<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a> This function gives the <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> of the <a href="/wiki/Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">multiplicative group of integers modulo n</a> (the <a href="/wiki/Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">group</a> of <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">units</a> of the <a href="/wiki/Ring_(algebra)" class="mw-redirect" title="Ring (algebra)">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /n\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /n\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2120ebbc85f91df66c6de5446367bf9fd620844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.658ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /n\mathbb {Z} }">).<a href="#cite_note-6">[6]</a> It is also used for defining the <a href="/wiki/RSA_(cryptosystem)" title="RSA (cryptosystem)">RSA encryption system</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History,_terminology,_and_notation">History, terminology, and notation</h2>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=1" title="Edit section: History, terminology, and notation">edit</a>]</div> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> introduced the function in 1763.<a href="#cite_note-7">[7]</a><a href="#cite_note-Sandifer,_p._203-8">[8]</a><a href="#cite_note-9">[9]</a> However, he did not at that time choose any specific symbol to denote it. In a 1784 publication, Euler studied the function further, choosing the Greek letter π to denote it: he wrote πD for "the multitude of numbers less than D, and which have no common divisor with it".<a href="#cite_note-10">[10]</a> This definition varies from the current definition for the totient function at D = 1 but is otherwise the same. The now-standard notation<a href="#cite_note-Sandifer,_p._203-8">[8]</a><a href="#cite_note-11">[11]</a> φ(A) comes from <a href="/wiki/Gauss" class="mw-redirect" title="Gauss">Gauss</a>'s 1801 treatise <a href="/wiki/Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a>,<a href="#cite_note-12">[12]</a><a href="#cite_note-13">[13]</a> although Gauss did not use parentheses around the argument and wrote φA. Thus, it is often called Euler's phi function or simply the phi function. In 1879, <a href="/wiki/James_Joseph_Sylvester" title="James Joseph Sylvester">J. J. Sylvester</a> coined the term totient for this function,<a href="#cite_note-14">[14]</a><a href="#cite_note-15">[15]</a> so it is also referred to as Euler's totient function, the Euler totient, or Euler's totient. <a href="/wiki/Jordan%27s_totient_function" title="Jordan's totient function">Jordan's totient</a> is a generalization of Euler's. The cototient of n is defined as n − φ(n). It counts the number of positive integers less than or equal to n that have at least one <a href="/wiki/Prime_number" title="Prime number">prime factor</a> in common with n. <div class="mw-heading mw-heading2"><h2 id="Computing_Euler's_totient_function">Computing Euler's totient function</h2>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=2" title="Edit section: Computing Euler's totient function">edit</a>]</div> There are several formulae for computing φ(n). <div class="mw-heading mw-heading3"><h3 id="Euler's_product_formula">Euler's product formula</h3>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=3" title="Edit section: Euler's product formula">edit</a>]</div> It states <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)=n\prod _{p\mid n}\left(1-{\frac {1}{p}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)=n\prod _{p\mid n}\left(1-{\frac {1}{p}}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6b6388ded7d1e160a3bd82b60c5b593947088a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:23.424ex; height:7.176ex;" alt="{\displaystyle \varphi (n)=n\prod _{p\mid n}\left(1-{\frac {1}{p}}\right),}"></dd></dl> where the product is over the distinct <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> dividing n. (For notation, see <a href="/wiki/Arithmetical_function#Notation" class="mw-redirect" title="Arithmetical function">Arithmetical function</a>.) An equivalent formulation is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)=p_{1}^{k_{1}-1}(p_{1}{-}1)\,p_{2}^{k_{2}-1}(p_{2}{-}1)\cdots p_{r}^{k_{r}-1}(p_{r}{-}1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)=p_{1}^{k_{1}-1}(p_{1}{-}1)\,p_{2}^{k_{2}-1}(p_{2}{-}1)\cdots p_{r}^{k_{r}-1}(p_{r}{-}1),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a2a86c67900e498a71aa14b7dd0179bc2069260" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.79ex; height:3.509ex;" alt="{\displaystyle \varphi (n)=p_{1}^{k_{1}-1}(p_{1}{-}1)\,p_{2}^{k_{2}-1}(p_{2}{-}1)\cdots p_{r}^{k_{r}-1}(p_{r}{-}1),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{r}^{k_{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{r}^{k_{r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f9538221d0fcae917b57da97b72ef3fca710ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.194ex; height:3.509ex;" alt="{\displaystyle n=p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{r}^{k_{r}}}"> is the <a href="/wiki/Prime_factorization" class="mw-redirect" title="Prime factorization">prime factorization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},p_{2},\ldots ,p_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1},p_{2},\ldots ,p_{r}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/534311dd876691edb53d98d5762a239435a82072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:12.892ex; height:2.009ex;" alt="{\displaystyle p_{1},p_{2},\ldots ,p_{r}}"> are distinct prime numbers). The proof of these formulae depends on two important facts. <div class="mw-heading mw-heading4"><h4 id="Phi_is_a_multiplicative_function">Phi is a multiplicative function</h4>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=4" title="Edit section: Phi is a multiplicative function">edit</a>]</div> This means that if gcd(m, n) = 1, then φ(m) φ(n) = φ(mn). Proof outline: Let A, B, C be the sets of positive integers which are <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> to and less than m, n, mn, respectively, so that |A| = φ(m), etc. Then there is a <a href="/wiki/Bijection" title="Bijection">bijection</a> between A × B and C by the <a href="/wiki/Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a>. <div class="mw-heading mw-heading4"><h4 id="Value_of_phi_for_a_prime_power_argument">Value of phi for a prime power argument</h4>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=5" title="Edit section: Value of phi for a prime power argument">edit</a>]</div> If p is prime and k ≥ 1, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \left(p^{k}\right)=p^{k}-p^{k-1}=p^{k-1}(p-1)=p^{k}\left(1-{\tfrac {1}{p}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mrow> <mo>(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \left(p^{k}\right)=p^{k}-p^{k-1}=p^{k-1}(p-1)=p^{k}\left(1-{\tfrac {1}{p}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a123bcd1c5c3214b80f4eeaa5d7f2e3e0700255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.509ex; height:4.843ex;" alt="{\displaystyle \varphi \left(p^{k}\right)=p^{k}-p^{k-1}=p^{k-1}(p-1)=p^{k}\left(1-{\tfrac {1}{p}}\right).}"></dd></dl> Proof: Since p is a prime number, the only possible values of gcd(pk, m) are 1, p, p2, ..., pk, and the only way to have gcd(pk, m) > 1 is if m is a multiple of p, that is, m ∈ {p, 2p, 3p, ..., pk − 1p = pk}, and there are pk − 1 such multiples not greater than pk. Therefore, the other pk − pk − 1 numbers are all relatively prime to pk. <div class="mw-heading mw-heading4"><h4 id="Proof_of_Euler's_product_formula">Proof of Euler's product formula</h4>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=6" title="Edit section: Proof of Euler's product formula">edit</a>]</div> The <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a> states that if n > 1 there is a unique expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{r}^{k_{r}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{r}^{k_{r}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24829d66bca81858264ebea940213d5fab1ca27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.841ex; height:3.509ex;" alt="{\displaystyle n=p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{r}^{k_{r}},}"> where p1 < p2 < ... < pr are <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> and each ki ≥ 1. (The case n = 1 corresponds to the empty product.) Repeatedly using the multiplicative property of φ and the formula for φ(pk) gives <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcl}\varphi (n)&=&\varphi (p_{1}^{k_{1}})\,\varphi (p_{2}^{k_{2}})\cdots \varphi (p_{r}^{k_{r}})\\[.1em]&=&p_{1}^{k_{1}}\left(1-{\frac {1}{p_{1}}}\right)p_{2}^{k_{2}}\left(1-{\frac {1}{p_{2}}}\right)\cdots p_{r}^{k_{r}}\left(1-{\frac {1}{p_{r}}}\right)\\[.1em]&=&p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{r}^{k_{r}}\left(1-{\frac {1}{p_{1}}}\right)\left(1-{\frac {1}{p_{2}}}\right)\cdots \left(1-{\frac {1}{p_{r}}}\right)\\[.1em]&=&n\left(1-{\frac {1}{p_{1}}}\right)\left(1-{\frac {1}{p_{2}}}\right)\cdots \left(1-{\frac {1}{p_{r}}}\right).\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left" rowspacing="0.5em 0.5em 0.5em 0.4em" columnspacing="1em"> <mtr> <mtd> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>φ</mi> <mo stretchy="false">(</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>φ</mi> <mo stretchy="false">(</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>⋯</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mi>n</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}\varphi (n)&=&\varphi (p_{1}^{k_{1}})\,\varphi (p_{2}^{k_{2}})\cdots \varphi (p_{r}^{k_{r}})\\[.1em]&=&p_{1}^{k_{1}}\left(1-{\frac {1}{p_{1}}}\right)p_{2}^{k_{2}}\left(1-{\frac {1}{p_{2}}}\right)\cdots p_{r}^{k_{r}}\left(1-{\frac {1}{p_{r}}}\right)\\[.1em]&=&p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{r}^{k_{r}}\left(1-{\frac {1}{p_{1}}}\right)\left(1-{\frac {1}{p_{2}}}\right)\cdots \left(1-{\frac {1}{p_{r}}}\right)\\[.1em]&=&n\left(1-{\frac {1}{p_{1}}}\right)\left(1-{\frac {1}{p_{2}}}\right)\cdots \left(1-{\frac {1}{p_{r}}}\right).\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7447918f84f4509322dda0b4aa4384ab8072a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:56.655ex; height:19.509ex;" alt="{\displaystyle {\begin{array}{rcl}\varphi (n)&=&\varphi (p_{1}^{k_{1}})\,\varphi (p_{2}^{k_{2}})\cdots \varphi (p_{r}^{k_{r}})\\[.1em]&=&p_{1}^{k_{1}}\left(1-{\frac {1}{p_{1}}}\right)p_{2}^{k_{2}}\left(1-{\frac {1}{p_{2}}}\right)\cdots p_{r}^{k_{r}}\left(1-{\frac {1}{p_{r}}}\right)\\[.1em]&=&p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{r}^{k_{r}}\left(1-{\frac {1}{p_{1}}}\right)\left(1-{\frac {1}{p_{2}}}\right)\cdots \left(1-{\frac {1}{p_{r}}}\right)\\[.1em]&=&n\left(1-{\frac {1}{p_{1}}}\right)\left(1-{\frac {1}{p_{2}}}\right)\cdots \left(1-{\frac {1}{p_{r}}}\right).\end{array}}}"></dd></dl> This gives both versions of Euler's product formula. An alternative proof that does not require the multiplicative property instead uses the <a href="/wiki/Inclusion-exclusion_principle" class="mw-redirect" title="Inclusion-exclusion principle">inclusion-exclusion principle</a> applied to the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots ,n\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ebfec86b3f22a18f086275390917d5aaa2d8c22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.257ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots ,n\}}">, excluding the sets of integers divisible by the prime divisors. <div class="mw-heading mw-heading4"><h4 id="Example">Example</h4>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=7" title="Edit section: Example">edit</a>]</div> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (20)=\varphi (2^{2}5)=20\,(1-{\tfrac {1}{2}})\,(1-{\tfrac {1}{5}})=20\cdot {\tfrac {1}{2}}\cdot {\tfrac {4}{5}}=8.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>20</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>5</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>20</mn> <mspace width="thinmathspace" /> <mo stretchy="false">(</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 stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>20</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>4</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>8.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (20)=\varphi (2^{2}5)=20\,(1-{\tfrac {1}{2}})\,(1-{\tfrac {1}{5}})=20\cdot {\tfrac {1}{2}}\cdot {\tfrac {4}{5}}=8.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9050fb7b46a26bcf6baab7e6051db0a8122258d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:53.605ex; height:3.676ex;" alt="{\displaystyle \varphi (20)=\varphi (2^{2}5)=20\,(1-{\tfrac {1}{2}})\,(1-{\tfrac {1}{5}})=20\cdot {\tfrac {1}{2}}\cdot {\tfrac {4}{5}}=8.}"></dd></dl> In words: the distinct prime factors of 20 are 2 and 5; half of the twenty integers from 1 to 20 are divisible by 2, leaving ten; a fifth of those are divisible by 5, leaving eight numbers coprime to 20; these are: 1, 3, 7, 9, 11, 13, 17, 19. The alternative formula uses only integers:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (20)=\varphi (2^{2}5^{1})=2^{2-1}(2{-}1)\,5^{1-1}(5{-}1)=2\cdot 1\cdot 1\cdot 4=8.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>20</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo>⋅</mo> <mn>4</mn> <mo>=</mo> <mn>8.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (20)=\varphi (2^{2}5^{1})=2^{2-1}(2{-}1)\,5^{1-1}(5{-}1)=2\cdot 1\cdot 1\cdot 4=8.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8047a8435e85e108c8ebbc39b46bf89fcb45378" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.214ex; height:3.176ex;" alt="{\displaystyle \varphi (20)=\varphi (2^{2}5^{1})=2^{2-1}(2{-}1)\,5^{1-1}(5{-}1)=2\cdot 1\cdot 1\cdot 4=8.}"> <div class="mw-heading mw-heading3"><h3 id="Fourier_transform">Fourier transform</h3>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=8" title="Edit section: Fourier transform">edit</a>]</div> The totient is the <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a> of the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">gcd</a>, evaluated at 1.<a href="#cite_note-16">[16]</a> Let <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\{\mathbf {x} \}[m]=\sum \limits _{k=1}^{n}x_{k}\cdot e^{{-2\pi i}{\frac {mk}{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{\mathbf {x} \}[m]=\sum \limits _{k=1}^{n}x_{k}\cdot e^{{-2\pi i}{\frac {mk}{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44406da8bf27ff396599d35e91d4328c3f961a81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.563ex; height:6.843ex;" alt="{\displaystyle {\mathcal {F}}\{\mathbf {x} \}[m]=\sum \limits _{k=1}^{n}x_{k}\cdot e^{{-2\pi i}{\frac {mk}{n}}}}"></dd></dl> where xk = gcd(k,n) for k ∈ {1, ..., n}. Then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)={\mathcal {F}}\{\mathbf {x} \}[1]=\sum \limits _{k=1}^{n}\gcd(k,n)e^{-2\pi i{\frac {k}{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>n</mi> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)={\mathcal {F}}\{\mathbf {x} \}[1]=\sum \limits _{k=1}^{n}\gcd(k,n)e^{-2\pi i{\frac {k}{n}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a88a09856ce4f73559b80fb01e44bdd70af8309" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.928ex; height:6.843ex;" alt="{\displaystyle \varphi (n)={\mathcal {F}}\{\mathbf {x} \}[1]=\sum \limits _{k=1}^{n}\gcd(k,n)e^{-2\pi i{\frac {k}{n}}}.}"></dd></dl> The real part of this formula is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)=\sum \limits _{k=1}^{n}\gcd(k,n)\cos {\tfrac {2\pi k}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)=\sum \limits _{k=1}^{n}\gcd(k,n)\cos {\tfrac {2\pi k}{n}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b7161db454cc257febbdbc867f677adac31177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.49ex; height:6.843ex;" alt="{\displaystyle \varphi (n)=\sum \limits _{k=1}^{n}\gcd(k,n)\cos {\tfrac {2\pi k}{n}}.}"></dd></dl> For example, using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\tfrac {\pi }{5}}={\tfrac {{\sqrt {5}}+1}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\tfrac {\pi }{5}}={\tfrac {{\sqrt {5}}+1}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd70ef072849c9b02aa5a881dda9687930ede301" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.502ex; height:4.343ex;" alt="{\displaystyle \cos {\tfrac {\pi }{5}}={\tfrac {{\sqrt {5}}+1}{4}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\tfrac {2\pi }{5}}={\tfrac {{\sqrt {5}}-1}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\tfrac {2\pi }{5}}={\tfrac {{\sqrt {5}}-1}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d35d3029792574ea52bd150b8b48fc6583088b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.324ex; height:4.343ex;" alt="{\displaystyle \cos {\tfrac {2\pi }{5}}={\tfrac {{\sqrt {5}}-1}{4}}}">:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcl}\varphi (10)&=&\gcd(1,10)\cos {\tfrac {2\pi }{10}}+\gcd(2,10)\cos {\tfrac {4\pi }{10}}+\gcd(3,10)\cos {\tfrac {6\pi }{10}}+\cdots +\gcd(10,10)\cos {\tfrac {20\pi }{10}}\\&=&1\cdot ({\tfrac {{\sqrt {5}}+1}{4}})+2\cdot ({\tfrac {{\sqrt {5}}-1}{4}})+1\cdot (-{\tfrac {{\sqrt {5}}-1}{4}})+2\cdot (-{\tfrac {{\sqrt {5}}+1}{4}})+5\cdot (-1)\\&&+\ 2\cdot (-{\tfrac {{\sqrt {5}}+1}{4}})+1\cdot (-{\tfrac {{\sqrt {5}}-1}{4}})+2\cdot ({\tfrac {{\sqrt {5}}-1}{4}})+1\cdot ({\tfrac {{\sqrt {5}}+1}{4}})+10\cdot (1)\\&=&4.\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>10</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mn>10</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>6</mn> <mi>π</mi> </mrow> <mn>10</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo>,</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>20</mn> <mi>π</mi> </mrow> <mn>10</mn> </mfrac> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mn>1</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>5</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mo>+</mo> <mtext> </mtext> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>10</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mn>4.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}\varphi (10)&=&\gcd(1,10)\cos {\tfrac {2\pi }{10}}+\gcd(2,10)\cos {\tfrac {4\pi }{10}}+\gcd(3,10)\cos {\tfrac {6\pi }{10}}+\cdots +\gcd(10,10)\cos {\tfrac {20\pi }{10}}\\&=&1\cdot ({\tfrac {{\sqrt {5}}+1}{4}})+2\cdot ({\tfrac {{\sqrt {5}}-1}{4}})+1\cdot (-{\tfrac {{\sqrt {5}}-1}{4}})+2\cdot (-{\tfrac {{\sqrt {5}}+1}{4}})+5\cdot (-1)\\&&+\ 2\cdot (-{\tfrac {{\sqrt {5}}+1}{4}})+1\cdot (-{\tfrac {{\sqrt {5}}-1}{4}})+2\cdot ({\tfrac {{\sqrt {5}}-1}{4}})+1\cdot ({\tfrac {{\sqrt {5}}+1}{4}})+10\cdot (1)\\&=&4.\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2346b5c022a38efdee490d21f1512172996bee3d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:94.141ex; height:16.176ex;" alt="{\displaystyle {\begin{array}{rcl}\varphi (10)&=&\gcd(1,10)\cos {\tfrac {2\pi }{10}}+\gcd(2,10)\cos {\tfrac {4\pi }{10}}+\gcd(3,10)\cos {\tfrac {6\pi }{10}}+\cdots +\gcd(10,10)\cos {\tfrac {20\pi }{10}}\\&=&1\cdot ({\tfrac {{\sqrt {5}}+1}{4}})+2\cdot ({\tfrac {{\sqrt {5}}-1}{4}})+1\cdot (-{\tfrac {{\sqrt {5}}-1}{4}})+2\cdot (-{\tfrac {{\sqrt {5}}+1}{4}})+5\cdot (-1)\\&&+\ 2\cdot (-{\tfrac {{\sqrt {5}}+1}{4}})+1\cdot (-{\tfrac {{\sqrt {5}}-1}{4}})+2\cdot ({\tfrac {{\sqrt {5}}-1}{4}})+1\cdot ({\tfrac {{\sqrt {5}}+1}{4}})+10\cdot (1)\\&=&4.\end{array}}}">Unlike the Euler product and the divisor sum formula, this one does not require knowing the factors of n. However, it does involve the calculation of the greatest common divisor of n and every positive integer less than n, which suffices to provide the factorization anyway. <div class="mw-heading mw-heading3"><h3 id="Divisor_sum">Divisor sum</h3>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=9" title="Edit section: Divisor sum">edit</a>]</div> The property established by Gauss,<a href="#cite_note-17">[17]</a> that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{d\mid n}\varphi (d)=n,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{d\mid n}\varphi (d)=n,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5704a6180e7a29fee9eeed4e7da4fa49c281b8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:13.427ex; height:6.009ex;" alt="{\displaystyle \sum _{d\mid n}\varphi (d)=n,}"></dd></dl> where the sum is over all positive divisors d of n, can be proven in several ways. (See <a href="/wiki/Arithmetical_function#Notation" class="mw-redirect" title="Arithmetical function">Arithmetical function</a> for notational conventions.) One proof is to note that φ(d) is also equal to the number of possible generators of the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> Cd ; specifically, if Cd = ⟨g⟩ with gd = 1, then gk is a generator for every k coprime to d. Since every element of Cn generates a cyclic <a href="/wiki/Subgroup" title="Subgroup">subgroup</a>, and each subgroup Cd ⊆ Cn is generated by precisely φ(d) elements of Cn, the formula follows.<a href="#cite_note-18">[18]</a> Equivalently, the formula can be derived by the same argument applied to the <a href="/wiki/Root_of_unity#Group_of_nth_roots_of_unity" title="Root of unity">multiplicative group of the nth roots of unity</a> and the <a href="/wiki/Primitive_root_of_unity" class="mw-redirect" title="Primitive root of unity">primitive dth roots of unity</a>. The formula can also be derived from elementary arithmetic.<a href="#cite_note-19">[19]</a> For example, let n = 20 and consider the positive fractions up to 1 with denominator 20: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{20}},\,{\tfrac {2}{20}},\,{\tfrac {3}{20}},\,{\tfrac {4}{20}},\,{\tfrac {5}{20}},\,{\tfrac {6}{20}},\,{\tfrac {7}{20}},\,{\tfrac {8}{20}},\,{\tfrac {9}{20}},\,{\tfrac {10}{20}},\,{\tfrac {11}{20}},\,{\tfrac {12}{20}},\,{\tfrac {13}{20}},\,{\tfrac {14}{20}},\,{\tfrac {15}{20}},\,{\tfrac {16}{20}},\,{\tfrac {17}{20}},\,{\tfrac {18}{20}},\,{\tfrac {19}{20}},\,{\tfrac {20}{20}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>6</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>7</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>8</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>9</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>10</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>11</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>12</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>13</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>14</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>15</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>16</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>17</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>18</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>19</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>20</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{20}},\,{\tfrac {2}{20}},\,{\tfrac {3}{20}},\,{\tfrac {4}{20}},\,{\tfrac {5}{20}},\,{\tfrac {6}{20}},\,{\tfrac {7}{20}},\,{\tfrac {8}{20}},\,{\tfrac {9}{20}},\,{\tfrac {10}{20}},\,{\tfrac {11}{20}},\,{\tfrac {12}{20}},\,{\tfrac {13}{20}},\,{\tfrac {14}{20}},\,{\tfrac {15}{20}},\,{\tfrac {16}{20}},\,{\tfrac {17}{20}},\,{\tfrac {18}{20}},\,{\tfrac {19}{20}},\,{\tfrac {20}{20}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e37e85e9a842dd016d657a1e56abc8c21cde048" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:77.248ex; height:3.843ex;" alt="{\displaystyle {\tfrac {1}{20}},\,{\tfrac {2}{20}},\,{\tfrac {3}{20}},\,{\tfrac {4}{20}},\,{\tfrac {5}{20}},\,{\tfrac {6}{20}},\,{\tfrac {7}{20}},\,{\tfrac {8}{20}},\,{\tfrac {9}{20}},\,{\tfrac {10}{20}},\,{\tfrac {11}{20}},\,{\tfrac {12}{20}},\,{\tfrac {13}{20}},\,{\tfrac {14}{20}},\,{\tfrac {15}{20}},\,{\tfrac {16}{20}},\,{\tfrac {17}{20}},\,{\tfrac {18}{20}},\,{\tfrac {19}{20}},\,{\tfrac {20}{20}}.}"></dd></dl> Put them into lowest terms: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{20}},\,{\tfrac {1}{10}},\,{\tfrac {3}{20}},\,{\tfrac {1}{5}},\,{\tfrac {1}{4}},\,{\tfrac {3}{10}},\,{\tfrac {7}{20}},\,{\tfrac {2}{5}},\,{\tfrac {9}{20}},\,{\tfrac {1}{2}},\,{\tfrac {11}{20}},\,{\tfrac {3}{5}},\,{\tfrac {13}{20}},\,{\tfrac {7}{10}},\,{\tfrac {3}{4}},\,{\tfrac {4}{5}},\,{\tfrac {17}{20}},\,{\tfrac {9}{10}},\,{\tfrac {19}{20}},\,{\tfrac {1}{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>10</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>7</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>9</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>11</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>13</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>7</mn> <mn>10</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>17</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>9</mn> <mn>10</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>19</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{20}},\,{\tfrac {1}{10}},\,{\tfrac {3}{20}},\,{\tfrac {1}{5}},\,{\tfrac {1}{4}},\,{\tfrac {3}{10}},\,{\tfrac {7}{20}},\,{\tfrac {2}{5}},\,{\tfrac {9}{20}},\,{\tfrac {1}{2}},\,{\tfrac {11}{20}},\,{\tfrac {3}{5}},\,{\tfrac {13}{20}},\,{\tfrac {7}{10}},\,{\tfrac {3}{4}},\,{\tfrac {4}{5}},\,{\tfrac {17}{20}},\,{\tfrac {9}{10}},\,{\tfrac {19}{20}},\,{\tfrac {1}{1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11220f898366625c2e6ec73891a7d188b00e2458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:70.026ex; height:3.843ex;" alt="{\displaystyle {\tfrac {1}{20}},\,{\tfrac {1}{10}},\,{\tfrac {3}{20}},\,{\tfrac {1}{5}},\,{\tfrac {1}{4}},\,{\tfrac {3}{10}},\,{\tfrac {7}{20}},\,{\tfrac {2}{5}},\,{\tfrac {9}{20}},\,{\tfrac {1}{2}},\,{\tfrac {11}{20}},\,{\tfrac {3}{5}},\,{\tfrac {13}{20}},\,{\tfrac {7}{10}},\,{\tfrac {3}{4}},\,{\tfrac {4}{5}},\,{\tfrac {17}{20}},\,{\tfrac {9}{10}},\,{\tfrac {19}{20}},\,{\tfrac {1}{1}}}"></dd></dl> These twenty fractions are all the positive <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>⁠k/d⁠ ≤ 1 whose denominators are the divisors d = 1, 2, 4, 5, 10, 20. The fractions with 20 as denominator are those with numerators relatively prime to 20, namely <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/20⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/20⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠7/20⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠9/20⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠11/20⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠13/20⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠17/20⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠19/20⁠; by definition this is φ(20) fractions. Similarly, there are φ(10) fractions with denominator 10, and φ(5) fractions with denominator 5, etc. Thus the set of twenty fractions is split into subsets of size φ(d) for each d dividing 20. A similar argument applies for any n. <a href="/wiki/M%C3%B6bius_inversion" class="mw-redirect" title="Möbius inversion">Möbius inversion</a> applied to the divisor sum formula gives <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)=\sum _{d\mid n}\mu \left(d\right)\cdot {\frac {n}{d}}=n\sum _{d\mid n}{\frac {\mu (d)}{d}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>μ</mi> <mrow> <mo>(</mo> <mi>d</mi> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>d</mi> </mfrac> </mrow> <mo>=</mo> <mi>n</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)=\sum _{d\mid n}\mu \left(d\right)\cdot {\frac {n}{d}}=n\sum _{d\mid n}{\frac {\mu (d)}{d}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22c03f0125cef739ded96195e0059a5a374f9b65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:34.821ex; height:7.343ex;" alt="{\displaystyle \varphi (n)=\sum _{d\mid n}\mu \left(d\right)\cdot {\frac {n}{d}}=n\sum _{d\mid n}{\frac {\mu (d)}{d}},}"></dd></dl> where μ is the <a href="/wiki/M%C3%B6bius_function" title="Möbius function">Möbius function</a>, the <a href="/wiki/Multiplicative_function" title="Multiplicative function">multiplicative function</a> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (p)=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (p)=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf6236076e559d85a495344c4295d29414949416" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.449ex; height:2.843ex;" alt="{\displaystyle \mu (p)=-1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (p^{k})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (p^{k})=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d0690dd8247ceb4713a5074b12f17886d020451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.73ex; height:3.176ex;" alt="{\displaystyle \mu (p^{k})=0}"> for each prime p and k ≥ 2. This formula may also be derived from the product formula by multiplying out <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \prod _{p\mid n}(1-{\frac {1}{p}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \prod _{p\mid n}(1-{\frac {1}{p}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a593ab2f9e824265547ce762739389288d3c5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.172ex; height:3.676ex;" alt="{\textstyle \prod _{p\mid n}(1-{\frac {1}{p}})}"> to get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{d\mid n}{\frac {\mu (d)}{d}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{d\mid n}{\frac {\mu (d)}{d}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2561daaf4eeb4fe72ca86ff14dec062a95fbbdea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.99ex; height:4.343ex;" alt="{\textstyle \sum _{d\mid n}{\frac {\mu (d)}{d}}.}"> An example:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varphi (20)&=\mu (1)\cdot 20+\mu (2)\cdot 10+\mu (4)\cdot 5+\mu (5)\cdot 4+\mu (10)\cdot 2+\mu (20)\cdot 1\\[.5em]&=1\cdot 20-1\cdot 10+0\cdot 5-1\cdot 4+1\cdot 2+0\cdot 1=8.\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="0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>20</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>20</mn> <mo>+</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>10</mn> <mo>+</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>5</mn> <mo>+</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>4</mn> <mo>+</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>2</mn> <mo>+</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mn>20</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mn>20</mn> <mo>−</mo> <mn>1</mn> <mo>⋅</mo> <mn>10</mn> <mo>+</mo> <mn>0</mn> <mo>⋅</mo> <mn>5</mn> <mo>−</mo> <mn>1</mn> <mo>⋅</mo> <mn>4</mn> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>+</mo> <mn>0</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>8.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varphi (20)&=\mu (1)\cdot 20+\mu (2)\cdot 10+\mu (4)\cdot 5+\mu (5)\cdot 4+\mu (10)\cdot 2+\mu (20)\cdot 1\\[.5em]&=1\cdot 20-1\cdot 10+0\cdot 5-1\cdot 4+1\cdot 2+0\cdot 1=8.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a4ce2d54734e410d4eea1d1a6b0d62ad7fbfed9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:71.646ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}\varphi (20)&=\mu (1)\cdot 20+\mu (2)\cdot 10+\mu (4)\cdot 5+\mu (5)\cdot 4+\mu (10)\cdot 2+\mu (20)\cdot 1\\[.5em]&=1\cdot 20-1\cdot 10+0\cdot 5-1\cdot 4+1\cdot 2+0\cdot 1=8.\end{aligned}}}"> <div class="mw-heading mw-heading2"><h2 id="Some_values">Some values</h2>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=10" title="Edit section: Some values">edit</a>]</div> The first 100 values (sequence <a href="//oeis.org/A000010" class="extiw" title="oeis:A000010">A000010</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) are shown in the table and graph below: <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:EulerPhi100.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/EulerPhi100.svg/220px-EulerPhi100.svg.png" decoding="async" width="220" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/EulerPhi100.svg/330px-EulerPhi100.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/EulerPhi100.svg/440px-EulerPhi100.svg.png 2x" data-file-width="703" data-file-height="521" /></a><figcaption>Graph of the first 100 values</figcaption></figure> <dl><dd><table class="wikitable" style="text-align: right"> <caption>φ(n) for 1 ≤ n ≤ 100 </caption> <tbody><tr> <th>+ </th> <th>1</th> <th>2</th> <th>3</th> <th>4</th> <th>5</th> <th>6</th> <th>7</th> <th>8</th> <th>9</th> <th>10 </th></tr> <tr> <th>0 </th> <td>1</td> <td>1</td> <td>2</td> <td>2</td> <td>4</td> <td>2</td> <td>6</td> <td>4</td> <td>6</td> <td>4 </td></tr> <tr> <th>10 </th> <td>10</td> <td>4</td> <td>12</td> <td>6</td> <td>8</td> <td>8</td> <td>16</td> <td>6</td> <td>18</td> <td>8 </td></tr> <tr> <th>20 </th> <td>12</td> <td>10</td> <td>22</td> <td>8</td> <td>20</td> <td>12</td> <td>18</td> <td>12</td> <td>28</td> <td>8 </td></tr> <tr> <th>30 </th> <td>30</td> <td>16</td> <td>20</td> <td>16</td> <td>24</td> <td>12</td> <td>36</td> <td>18</td> <td>24</td> <td>16 </td></tr> <tr> <th>40 </th> <td>40</td> <td>12</td> <td>42</td> <td>20</td> <td>24</td> <td>22</td> <td>46</td> <td>16</td> <td>42</td> <td>20 </td></tr> <tr> <th>50 </th> <td>32</td> <td>24</td> <td>52</td> <td>18</td> <td>40</td> <td>24</td> <td>36</td> <td>28</td> <td>58</td> <td>16 </td></tr> <tr> <th>60 </th> <td>60</td> <td>30</td> <td>36</td> <td>32</td> <td>48</td> <td>20</td> <td>66</td> <td>32</td> <td>44</td> <td>24 </td></tr> <tr> <th>70 </th> <td>70</td> <td>24</td> <td>72</td> <td>36</td> <td>40</td> <td>36</td> <td>60</td> <td>24</td> <td>78</td> <td>32 </td></tr> <tr> <th>80 </th> <td>54</td> <td>40</td> <td>82</td> <td>24</td> <td>64</td> <td>42</td> <td>56</td> <td>40</td> <td>88</td> <td>24 </td></tr> <tr> <th>90 </th> <td>72</td> <td>44</td> <td>60</td> <td>46</td> <td>72</td> <td>32</td> <td>96</td> <td>42</td> <td>60</td> <td>40 </td></tr></tbody></table></dd></dl> In the graph at right the top line y = n − 1 is an <a href="/wiki/Upper_bound" class="mw-redirect" title="Upper bound">upper bound</a> valid for all n other than one, and attained if and only if n is a prime number. A simple lower bound is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)\geq {\sqrt {n/2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)\geq {\sqrt {n/2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60aa32e1775a3004aa91a683ca28b73ffd150306" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.866ex; height:4.843ex;" alt="{\displaystyle \varphi (n)\geq {\sqrt {n/2}}}">, which is rather loose: in fact, the <a href="/wiki/Limit_superior_and_limit_inferior" class="mw-redirect" title="Limit superior and limit inferior">lower limit</a> of the graph is proportional to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠n/log log n⁠.<a href="#cite_note-hw328-20">[20]</a> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Euler's_theorem">Euler's theorem</h2>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=11" title="Edit section: Euler's theorem">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/Euler%27s_theorem" title="Euler's theorem">Euler's theorem</a></div> This states that if a and n are <a href="/wiki/Relatively_prime" class="mw-redirect" title="Relatively prime">relatively prime</a> then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{\varphi (n)}\equiv 1\mod n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\varphi (n)}\equiv 1\mod n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5590d33336e9f15ac4d7592ca23e074f440a1051" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.367ex; height:2.843ex;" alt="{\displaystyle a^{\varphi (n)}\equiv 1\mod n.}"></dd></dl> The special case where n is prime is known as <a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a>. This follows from <a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a> and the fact that φ(n) is the <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> of the <a href="/wiki/Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">multiplicative group of integers modulo n</a>. The <a href="/wiki/RSA_(algorithm)" class="mw-redirect" title="RSA (algorithm)">RSA cryptosystem</a> is based on this theorem: it implies that the <a href="/wiki/Inverse_function" title="Inverse function">inverse</a> of the function a ↦ ae mod n, where e is the (public) encryption exponent, is the function b ↦ bd mod n, where d, the (private) decryption exponent, is the <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> of e modulo φ(n). The difficulty of computing φ(n) without knowing the factorization of n is thus the difficulty of computing d: this is known as the <a href="/wiki/RSA_problem" title="RSA problem">RSA problem</a> which can be solved by factoring n. The owner of the private key knows the factorization, since an RSA private key is constructed by choosing n as the product of two (randomly chosen) large primes p and q. Only n is publicly disclosed, and given the <a href="/wiki/Integer_factorization" title="Integer factorization">difficulty to factor large numbers</a> we have the guarantee that no one else knows the factorization. <div class="mw-heading mw-heading2"><h2 id="Other_formulae">Other formulae</h2>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=12" title="Edit section: Other formulae">edit</a>]</div> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mid b\implies \varphi (a)\mid \varphi (b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∣</mo> <mi>b</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mid b\implies \varphi (a)\mid \varphi (b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f87b161831921ef03a45e796d94fe903da8b6082" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.374ex; height:2.843ex;" alt="{\displaystyle a\mid b\implies \varphi (a)\mid \varphi (b)}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\mid \varphi (a^{m}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∣</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\mid \varphi (a^{m}-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5258fffd25e4ab850c1b5eb7cf13dc52f7ed63b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.215ex; height:2.843ex;" alt="{\displaystyle m\mid \varphi (a^{m}-1)}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (mn)=\varphi (m)\varphi (n)\cdot {\frac {d}{\varphi (d)}}\quad {\text{where }}d=\operatorname {gcd} (m,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>where </mtext> </mrow> <mi>d</mi> <mo>=</mo> <mi>gcd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (mn)=\varphi (m)\varphi (n)\cdot {\frac {d}{\varphi (d)}}\quad {\text{where }}d=\operatorname {gcd} (m,n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3a395aedf638cf2903002d51598c8c1f477c52f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.948ex; height:6.176ex;" alt="{\displaystyle \varphi (mn)=\varphi (m)\varphi (n)\cdot {\frac {d}{\varphi (d)}}\quad {\text{where }}d=\operatorname {gcd} (m,n)}"> In particular: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (2m)={\begin{cases}2\varphi (m)&{\text{ if }}m{\text{ is even}}\\\varphi (m)&{\text{ if }}m{\text{ is odd}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>2</mn> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is even</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is odd</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (2m)={\begin{cases}2\varphi (m)&{\text{ if }}m{\text{ is even}}\\\varphi (m)&{\text{ if }}m{\text{ is odd}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69a4a6f67d705dd054657cef3d1fba6882d5dfc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.85ex; height:6.176ex;" alt="{\displaystyle \varphi (2m)={\begin{cases}2\varphi (m)&{\text{ if }}m{\text{ is even}}\\\varphi (m)&{\text{ if }}m{\text{ is odd}}\end{cases}}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \left(n^{m}\right)=n^{m-1}\varphi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mrow> <mo>(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \left(n^{m}\right)=n^{m-1}\varphi (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982dff02a451ef8353a9d17f2b346f7d83bbc900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.779ex; height:3.176ex;" alt="{\displaystyle \varphi \left(n^{m}\right)=n^{m-1}\varphi (n)}"></li></ul> </li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\operatorname {lcm} (m,n))\cdot \varphi (\operatorname {gcd} (m,n))=\varphi (m)\cdot \varphi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>lcm</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>gcd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (\operatorname {lcm} (m,n))\cdot \varphi (\operatorname {gcd} (m,n))=\varphi (m)\cdot \varphi (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90c2e2f7938febf6d3aea0bb24bb45416c45e6e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.869ex; height:2.843ex;" alt="{\displaystyle \varphi (\operatorname {lcm} (m,n))\cdot \varphi (\operatorname {gcd} (m,n))=\varphi (m)\cdot \varphi (n)}"> Compare this to the formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {lcm} (m,n)\cdot \operatorname {gcd} (m,n)=m\cdot n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>lcm</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>gcd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> <mo>⋅</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {lcm} (m,n)\cdot \operatorname {gcd} (m,n)=m\cdot n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/415bc6f0929577fcf25d3f465d0161c0d714eeeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.551ex; height:2.843ex;" alt="{\textstyle \operatorname {lcm} (m,n)\cdot \operatorname {gcd} (m,n)=m\cdot n}"> (see <a href="/wiki/Least_common_multiple" title="Least common multiple">least common multiple</a>). </li> <li>φ(n) is even for n ≥ 3. Moreover, if n has r distinct odd prime factors, 2r | φ(n)</li> <li> For any a > 1 and n > 6 such that 4 ∤ n there exists an l ≥ 2n such that l | φ(an − 1).</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\varphi (n)}{n}}={\frac {\varphi (\operatorname {rad} (n))}{\operatorname {rad} (n)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>rad</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>rad</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\varphi (n)}{n}}={\frac {\varphi (\operatorname {rad} (n))}{\operatorname {rad} (n)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d43e5a214d351a784804c49c708f93a64dfd8d93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.395ex; height:6.509ex;" alt="{\displaystyle {\frac {\varphi (n)}{n}}={\frac {\varphi (\operatorname {rad} (n))}{\operatorname {rad} (n)}}}"> where rad(n) is the <a href="/wiki/Radical_of_an_integer" title="Radical of an integer">radical of n</a> (the product of all distinct primes dividing <a href="/wiki/Radical_of_an_integer" title="Radical of an integer">n</a>).</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{d\mid n}{\frac {\mu ^{2}(d)}{\varphi (d)}}={\frac {n}{\varphi (n)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{d\mid n}{\frac {\mu ^{2}(d)}{\varphi (d)}}={\frac {n}{\varphi (n)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a910cb8792ad63d84754b85bbf7beeb19c2c43f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:18.718ex; height:7.509ex;" alt="{\displaystyle \sum _{d\mid n}{\frac {\mu ^{2}(d)}{\varphi (d)}}={\frac {n}{\varphi (n)}}}"> <a href="#cite_note-21">[21]</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{1\leq k\leq n-1 \atop gcd(k,n)=1}\!\!k={\tfrac {1}{2}}n\varphi (n)\quad {\text{for }}n>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>g</mi> <mi>c</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </munder> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>n</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>n</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{1\leq k\leq n-1 \atop gcd(k,n)=1}\!\!k={\tfrac {1}{2}}n\varphi (n)\quad {\text{for }}n>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06b83f957f11ec754a24e79f1eb2f5cd365a92c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:30.13ex; height:7.343ex;" alt="{\displaystyle \sum _{1\leq k\leq n-1 \atop gcd(k,n)=1}\!\!k={\tfrac {1}{2}}n\varphi (n)\quad {\text{for }}n>1}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}\varphi (k)={\tfrac {1}{2}}\left(1+\sum _{k=1}^{n}\mu (k)\left\lfloor {\frac {n}{k}}\right\rfloor ^{2}\right)={\frac {3}{\pi ^{2}}}n^{2}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <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>n</mi> </mrow> </munderover> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <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> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}\varphi (k)={\tfrac {1}{2}}\left(1+\sum _{k=1}^{n}\mu (k)\left\lfloor {\frac {n}{k}}\right\rfloor ^{2}\right)={\frac {3}{\pi ^{2}}}n^{2}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1013b804a82e91ba6cafe2cd6ab0c1c43d591d1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:73.56ex; height:7.509ex;" alt="{\displaystyle \sum _{k=1}^{n}\varphi (k)={\tfrac {1}{2}}\left(1+\sum _{k=1}^{n}\mu (k)\left\lfloor {\frac {n}{k}}\right\rfloor ^{2}\right)={\frac {3}{\pi ^{2}}}n^{2}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)}"> (<a href="#cite_note-Wal1963-22">[22]</a> cited in<a href="#cite_note-23">[23]</a>)</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}\varphi (k)={\frac {3}{\pi ^{2}}}n^{2}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {1}{3}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <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> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}\varphi (k)={\frac {3}{\pi ^{2}}}n^{2}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {1}{3}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd484521b084cfae8554c6a1501c97e477176a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.827ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}\varphi (k)={\frac {3}{\pi ^{2}}}n^{2}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {1}{3}}\right)}"> [Liu (2016)] </li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}{\frac {\varphi (k)}{k}}=\sum _{k=1}^{n}{\frac {\mu (k)}{k}}\left\lfloor {\frac {n}{k}}\right\rfloor ={\frac {6}{\pi ^{2}}}n+O\left((\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </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>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </mfrac> </mrow> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>n</mi> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <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> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}{\frac {\varphi (k)}{k}}=\sum _{k=1}^{n}{\frac {\mu (k)}{k}}\left\lfloor {\frac {n}{k}}\right\rfloor ={\frac {6}{\pi ^{2}}}n+O\left((\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/771ae00bb94d7b7620659ac090015dea93fd57bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:62.386ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}{\frac {\varphi (k)}{k}}=\sum _{k=1}^{n}{\frac {\mu (k)}{k}}\left\lfloor {\frac {n}{k}}\right\rfloor ={\frac {6}{\pi ^{2}}}n+O\left((\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)}"> <a href="#cite_note-Wal1963-22">[22]</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}{\frac {k}{\varphi (k)}}={\frac {315\,\zeta (3)}{2\pi ^{4}}}n-{\frac {\log n}{2}}+O\left((\log n)^{\frac {2}{3}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>315</mn> <mspace width="thinmathspace" /> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}{\frac {k}{\varphi (k)}}={\frac {315\,\zeta (3)}{2\pi ^{4}}}n-{\frac {\log n}{2}}+O\left((\log n)^{\frac {2}{3}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d61e0dd1e45d522a51ca916a4cb606af07136c30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.54ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}{\frac {k}{\varphi (k)}}={\frac {315\,\zeta (3)}{2\pi ^{4}}}n-{\frac {\log n}{2}}+O\left((\log n)^{\frac {2}{3}}\right)}"> <a href="#cite_note-Sita-24">[24]</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}{\frac {1}{\varphi (k)}}={\frac {315\,\zeta (3)}{2\pi ^{4}}}\left(\log n+\gamma -\sum _{p{\text{ prime}}}{\frac {\log p}{p^{2}-p+1}}\right)+O\left({\frac {(\log n)^{\frac {2}{3}}}{n}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>315</mn> <mspace width="thinmathspace" /> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo>+</mo> <mi>γ</mi> <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> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>p</mi> </mrow> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}{\frac {1}{\varphi (k)}}={\frac {315\,\zeta (3)}{2\pi ^{4}}}\left(\log n+\gamma -\sum _{p{\text{ prime}}}{\frac {\log p}{p^{2}-p+1}}\right)+O\left({\frac {(\log n)^{\frac {2}{3}}}{n}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27625122446f6d1d4eb2ddd05c18991266037d2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:71.944ex; height:8.843ex;" alt="{\displaystyle \sum _{k=1}^{n}{\frac {1}{\varphi (k)}}={\frac {315\,\zeta (3)}{2\pi ^{4}}}\left(\log n+\gamma -\sum _{p{\text{ prime}}}{\frac {\log p}{p^{2}-p+1}}\right)+O\left({\frac {(\log n)^{\frac {2}{3}}}{n}}\right)}"> <a href="#cite_note-Sita-24">[24]</a>(where γ is the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>).</li> </ul> <div class="mw-heading mw-heading3"><h3 id="Menon's_identity">Menon's identity</h3>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=13" title="Edit section: Menon's identity">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/Arithmetic_function#Menon.27s_identity" title="Arithmetic function">Menon's identity</a></div> In 1965 P. Kesava Menon proved <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\!\!\!\!\gcd(k-1,n)=\varphi (n)d(n),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>n</mi> </mrow> </mover> </mrow> </mrow> </munder> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\!\!\!\!\gcd(k-1,n)=\varphi (n)d(n),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7c271a3cfbc0cb0ecfb8c9c17465d8fa7919e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:32.813ex; height:7.676ex;" alt="{\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\!\!\!\!\gcd(k-1,n)=\varphi (n)d(n),}"></dd></dl> where <a href="/wiki/Divisor_function" title="Divisor function">d(n) = σ0(n)</a> is the number of divisors of n. <div class="mw-heading mw-heading3"><h3 id="Divisibility_by_any_fixed_positive_integer">Divisibility by any fixed positive integer</h3>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=14" title="Edit section: Divisibility by any fixed positive integer">edit</a>]</div> The following property, which is part of the « folklore » (i.e., apparently unpublished as a specific result:<a href="#cite_note-25">[25]</a> see the introduction of this article in which it is stated as having « long been known ») has important consequences. For instance it rules out uniform distribution of the values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f067864064667dd5f8b2508b9cbf983d89788629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle \varphi (n)}"> in the arithmetic progressions modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> for any integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe3606e1cf4480eb39e5ddcd46d4dae2067c0b5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q>1}">. <ul><li>For every fixed positive integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}">, the relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q|\varphi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q|\varphi (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a37ffa257e3083809aaf1aad4869db5e59908d75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.441ex; height:2.843ex;" alt="{\displaystyle q|\varphi (n)}"> holds for almost all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, meaning for all but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/893488c26042998c6368d70fd28334ba7ccfc7f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.267ex; height:2.843ex;" alt="{\displaystyle o(x)}"> values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\leq x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\leq x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/629e475fed09acb4b4b49c94d9ea781619ade867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.823ex; height:2.176ex;" alt="{\displaystyle n\leq x}"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\rightarrow \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02579e74e2ef1ca0befceba816b311fe5bfd6844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.268ex; height:1.843ex;" alt="{\displaystyle x\rightarrow \infty }">.</li></ul> This is an elementary consequence of the fact that the sum of the reciprocals of the primes congruent to 1 modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> diverges, which itself is a corollary of the proof of <a href="/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions" title="Dirichlet's theorem on arithmetic progressions">Dirichlet's theorem on arithmetic progressions</a>. <div class="mw-heading mw-heading2"><h2 id="Generating_functions">Generating functions</h2>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=15" title="Edit section: Generating functions">edit</a>]</div> The <a href="/wiki/Dirichlet_series" title="Dirichlet series">Dirichlet series</a> for φ(n) may be written in terms of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> as:<a href="#cite_note-26">[26]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\frac {\varphi (n)}{n^{s}}}={\frac {\zeta (s-1)}{\zeta (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> <mrow> <mi>φ</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> <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>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {\varphi (n)}{n^{s}}}={\frac {\zeta (s-1)}{\zeta (s)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/503a149acd9f861d3db81875ce98fcb458afbf9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.235ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {\varphi (n)}{n^{s}}}={\frac {\zeta (s-1)}{\zeta (s)}}}"></dd></dl> where the left-hand side converges for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (s)>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">ℜ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Re (s)>2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f75c84c17f43158d1f7fc4c3e8ece0e0bb3f8140" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \Re (s)>2}">. The <a href="/wiki/Lambert_series" title="Lambert series">Lambert series</a> generating function is<a href="#cite_note-27">[27]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\frac {\varphi (n)q^{n}}{1-q^{n}}}={\frac {q}{(1-q)^{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>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {\varphi (n)q^{n}}{1-q^{n}}}={\frac {q}{(1-q)^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/995c90fce0364c2acd4bc66fd92bab5683eb1ff5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.471ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {\varphi (n)q^{n}}{1-q^{n}}}={\frac {q}{(1-q)^{2}}}}"></dd></dl> which converges for |q| < 1. Both of these are proved by elementary series manipulations and the formulae for φ(n). <div class="mw-heading mw-heading2"><h2 id="Growth_rate">Growth rate</h2>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=16" title="Edit section: Growth rate">edit</a>]</div> In the words of Hardy & Wright, the order of φ(n) is "always 'nearly n'."<a href="#cite_note-28">[28]</a> First<a href="#cite_note-29">[29]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim \sup {\frac {\varphi (n)}{n}}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">lim</mo> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim \sup {\frac {\varphi (n)}{n}}=1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae70ed67dfdbe2c8653cdff5e9d78a1ec90cd55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.973ex; height:5.676ex;" alt="{\displaystyle \lim \sup {\frac {\varphi (n)}{n}}=1,}"></dd></dl> but as n goes to infinity,<a href="#cite_note-30">[30]</a> for all δ > 0 <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\varphi (n)}{n^{1-\delta }}}\rightarrow \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>δ</mi> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\varphi (n)}{n^{1-\delta }}}\rightarrow \infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e1e00f421006da595cde383bc5d8af3d089196c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.145ex; height:6.009ex;" alt="{\displaystyle {\frac {\varphi (n)}{n^{1-\delta }}}\rightarrow \infty .}"></dd></dl> These two formulae can be proved by using little more than the formulae for φ(n) and the <a href="/wiki/Divisor_function" title="Divisor function">divisor sum function</a> σ(n). In fact, during the proof of the second formula, the inequality <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {6}{\pi ^{2}}}<{\frac {\varphi (n)\sigma (n)}{n^{2}}}<1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo><</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {6}{\pi ^{2}}}<{\frac {\varphi (n)\sigma (n)}{n^{2}}}<1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6499edd9d3d2f8a3ae19afa85aae7e349d08fae5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.325ex; height:6.009ex;" alt="{\displaystyle {\frac {6}{\pi ^{2}}}<{\frac {\varphi (n)\sigma (n)}{n^{2}}}<1,}"></dd></dl> true for n > 1, is proved. We also have<a href="#cite_note-hw328-20">[20]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim \inf {\frac {\varphi (n)}{n}}\log \log n=e^{-\gamma }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">lim</mo> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim \inf {\frac {\varphi (n)}{n}}\log \log n=e^{-\gamma }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae055282dae94c3579d0c12e0bbbba4f24d2d0ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.947ex; height:5.676ex;" alt="{\displaystyle \lim \inf {\frac {\varphi (n)}{n}}\log \log n=e^{-\gamma }.}"></dd></dl> Here γ is <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler's constant</a>, γ = 0.577215665..., so eγ = 1.7810724... and e−γ = 0.56145948.... Proving this does not quite require the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a>.<a href="#cite_note-31">[31]</a><a href="#cite_note-32">[32]</a> Since log log n goes to infinity, this formula shows that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim \inf {\frac {\varphi (n)}{n}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">lim</mo> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim \inf {\frac {\varphi (n)}{n}}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca38585ab052056e9f1783c20579a84a5444d5fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.123ex; height:5.676ex;" alt="{\displaystyle \lim \inf {\frac {\varphi (n)}{n}}=0.}"></dd></dl> In fact, more is true.<a href="#cite_note-33">[33]</a><a href="#cite_note-34">[34]</a><a href="#cite_note-Rib320-35">[35]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)>{\frac {n}{e^{\gamma }\;\log \log n+{\frac {3}{\log \log n}}}}\quad {\text{for }}n>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mspace width="thickmathspace" /> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>n</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)>{\frac {n}{e^{\gamma }\;\log \log n+{\frac {3}{\log \log n}}}}\quad {\text{for }}n>2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54a7df0105c4627f91ce307dfe4fbeee1a065a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:40.996ex; height:6.676ex;" alt="{\displaystyle \varphi (n)>{\frac {n}{e^{\gamma }\;\log \log n+{\frac {3}{\log \log n}}}}\quad {\text{for }}n>2}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)<{\frac {n}{e^{\gamma }\log \log n}}\quad {\text{for infinitely many }}n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for infinitely many </mtext> </mrow> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)<{\frac {n}{e^{\gamma }\log \log n}}\quad {\text{for infinitely many }}n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99a466e8c7a7e709529c358f0891f6afe18f50c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.926ex; height:5.176ex;" alt="{\displaystyle \varphi (n)<{\frac {n}{e^{\gamma }\log \log n}}\quad {\text{for infinitely many }}n.}"></dd></dl> The second inequality was shown by <a href="/wiki/Jean-Louis_Nicolas" title="Jean-Louis Nicolas">Jean-Louis Nicolas</a>. Ribenboim says "The method of proof is interesting, in that the inequality is shown first under the assumption that the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> is true, secondly under the contrary assumption."<a href="#cite_note-Rib320-35">[35]</a>: 173  For the average order, we have<a href="#cite_note-Wal1963-22">[22]</a><a href="#cite_note-SMC2425-36">[36]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (1)+\varphi (2)+\cdots +\varphi (n)={\frac {3n^{2}}{\pi ^{2}}}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)\quad {\text{as }}n\rightarrow \infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <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> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>as </mtext> </mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (1)+\varphi (2)+\cdots +\varphi (n)={\frac {3n^{2}}{\pi ^{2}}}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)\quad {\text{as }}n\rightarrow \infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7850d2f689e3d7a105765e63bd07c55cf543fe27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:75.619ex; height:6.343ex;" alt="{\displaystyle \varphi (1)+\varphi (2)+\cdots +\varphi (n)={\frac {3n^{2}}{\pi ^{2}}}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)\quad {\text{as }}n\rightarrow \infty ,}"></dd></dl> due to <a href="/wiki/Arnold_Walfisz" title="Arnold Walfisz">Arnold Walfisz</a>, its proof exploiting estimates on exponential sums due to <a href="/wiki/Ivan_Matveevich_Vinogradov" class="mw-redirect" title="Ivan Matveevich Vinogradov">I. M. Vinogradov</a> and <a href="/wiki/N._M._Korobov" class="mw-redirect" title="N. M. Korobov">N. M. Korobov</a>. By a combination of van der Corput's and Vinogradov's methods, H.-Q. Liu (On Euler's function.Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 4, 769–775) improved the error term to <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {1}{3}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <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> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {1}{3}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/154cbacb8bd69d2f4df18a1f498918f74377214b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.932ex; height:6.176ex;" alt="{\displaystyle O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {1}{3}}\right)}"></dd></dl> (this is currently the best known estimate of this type). The <a href="/wiki/Big_O_notation" title="Big O notation">"Big O"</a> stands for a quantity that is bounded by a constant times the function of n inside the parentheses (which is small compared to n2). This result can be used to prove<a href="#cite_note-37">[37]</a> that <a href="/wiki/Coprime_integers#Probability_of_coprimality" title="Coprime integers">the probability of two randomly chosen numbers being relatively prime</a> is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠6/π2⁠. <div class="mw-heading mw-heading2"><h2 id="Ratio_of_consecutive_values">Ratio of consecutive values</h2>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=17" title="Edit section: Ratio of consecutive values">edit</a>]</div> In 1950 Somayajulu proved<a href="#cite_note-Rib38-38">[38]</a><a href="#cite_note-SMC16-39">[39]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\lim \inf {\frac {\varphi (n+1)}{\varphi (n)}}&=0\quad {\text{and}}\\[5px]\lim \sup {\frac {\varphi (n+1)}{\varphi (n)}}&=\infty .\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="0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo movablelimits="true" form="prefix">lim</mo> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo movablelimits="true" form="prefix">lim</mo> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\lim \inf {\frac {\varphi (n+1)}{\varphi (n)}}&=0\quad {\text{and}}\\[5px]\lim \sup {\frac {\varphi (n+1)}{\varphi (n)}}&=\infty .\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc005e44784aa197220263d20f9a495ee655b528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:28.151ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\lim \inf {\frac {\varphi (n+1)}{\varphi (n)}}&=0\quad {\text{and}}\\[5px]\lim \sup {\frac {\varphi (n+1)}{\varphi (n)}}&=\infty .\end{aligned}}}"></dd></dl> In 1954 <a href="/wiki/Andrzej_Schinzel" title="Andrzej Schinzel">Schinzel</a> and <a href="/wiki/Wac%C5%82aw_Sierpi%C5%84ski" title="Wacław Sierpiński">Sierpiński</a> strengthened this, proving<a href="#cite_note-Rib38-38">[38]</a><a href="#cite_note-SMC16-39">[39]</a> that the set <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\frac {\varphi (n+1)}{\varphi (n)}},\;\;n=1,2,\ldots \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\frac {\varphi (n+1)}{\varphi (n)}},\;\;n=1,2,\ldots \right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c9cecc224f1d9d7312c6cd97006a15fb8f51bef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.983ex; height:6.509ex;" alt="{\displaystyle \left\{{\frac {\varphi (n+1)}{\varphi (n)}},\;\;n=1,2,\ldots \right\}}"></dd></dl> is <a href="/wiki/Dense_set" title="Dense set">dense</a> in the positive real numbers. They also proved<a href="#cite_note-Rib38-38">[38]</a> that the set <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\frac {\varphi (n)}{n}},\;\;n=1,2,\ldots \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\frac {\varphi (n)}{n}},\;\;n=1,2,\ldots \right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93cab0a62bd81ed29dfa211d862b73529eb8ddc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.98ex; height:6.343ex;" alt="{\displaystyle \left\{{\frac {\varphi (n)}{n}},\;\;n=1,2,\ldots \right\}}"></dd></dl> is dense in the interval (0,1). <div class="mw-heading mw-heading2"><h2 id="Totient_numbers">Totient numbers</h2>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=18" title="Edit section: Totient numbers">edit</a>]</div> A totient number is a value of Euler's totient function: that is, an m for which there is at least one n for which φ(n) = m. The valency or multiplicity of a totient number m is the number of solutions to this equation.<a href="#cite_note-Guy144-40">[40]</a> A <a href="/wiki/Nontotient" title="Nontotient">nontotient</a> is a natural number which is not a totient number. Every odd integer exceeding 1 is trivially a nontotient. There are also infinitely many even nontotients,<a href="#cite_note-SC230-41">[41]</a> and indeed every positive integer has a multiple which is an even nontotient.<a href="#cite_note-Zha1993-42">[42]</a> The number of totient numbers up to a given limit x is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x}{\log x}}e^{{\big (}C+o(1){\big )}(\log \log \log x)^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>C</mi> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{\log x}}e^{{\big (}C+o(1){\big )}(\log \log \log x)^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83eb2835d46f2734d1a1de492b70eb731b1591cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.746ex; height:5.843ex;" alt="{\displaystyle {\frac {x}{\log x}}e^{{\big (}C+o(1){\big )}(\log \log \log x)^{2}}}"></dd></dl> for a constant C = 0.8178146....<a href="#cite_note-Ford1998-43">[43]</a> If counted accordingly to multiplicity, the number of totient numbers up to a given limit x is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Big \vert }\{n:\varphi (n)\leq x\}{\Big \vert }={\frac {\zeta (2)\zeta (3)}{\zeta (6)}}\cdot x+R(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>:</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi>x</mi> <mo>+</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Big \vert }\{n:\varphi (n)\leq x\}{\Big \vert }={\frac {\zeta (2)\zeta (3)}{\zeta (6)}}\cdot x+R(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1c25e83be96c10ab760e1badf4924ec3429b902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.923ex; height:6.509ex;" alt="{\displaystyle {\Big \vert }\{n:\varphi (n)\leq x\}{\Big \vert }={\frac {\zeta (2)\zeta (3)}{\zeta (6)}}\cdot x+R(x)}"></dd></dl> where the error term R is of order at most <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠x/(log x)k⁠ for any positive k.<a href="#cite_note-SMC22-44">[44]</a> It is known that the multiplicity of m exceeds mδ infinitely often for any δ < 0.55655.<a href="#cite_note-SMC21-45">[45]</a><a href="#cite_note-Guy145-46">[46]</a> <div class="mw-heading mw-heading3"><h3 id="Ford's_theorem">Ford's theorem</h3>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=19" title="Edit section: Ford's theorem">edit</a>]</div> <a href="#CITEREFFord1999">Ford (1999)</a> proved that for every integer k ≥ 2 there is a totient number m of multiplicity k: that is, for which the equation φ(n) = m has exactly k solutions; this result had previously been conjectured by <a href="/wiki/Wac%C5%82aw_Sierpi%C5%84ski" title="Wacław Sierpiński">Wacław Sierpiński</a>,<a href="#cite_note-SC229-47">[47]</a> and it had been obtained as a consequence of <a href="/wiki/Schinzel%27s_hypothesis_H" title="Schinzel's hypothesis H">Schinzel's hypothesis H</a>.<a href="#cite_note-Ford1998-43">[43]</a> Indeed, each multiplicity that occurs, does so infinitely often.<a href="#cite_note-Ford1998-43">[43]</a><a href="#cite_note-Guy145-46">[46]</a> However, no number m is known with multiplicity k = 1. <a href="/wiki/Carmichael%27s_totient_function_conjecture" title="Carmichael's totient function conjecture">Carmichael's totient function conjecture</a> is the statement that there is no such m.<a href="#cite_note-SC228-48">[48]</a> <div class="mw-heading mw-heading3"><h3 id="Perfect_totient_numbers">Perfect totient numbers</h3>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=20" title="Edit section: Perfect totient numbers">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/Perfect_totient_number" title="Perfect totient number">Perfect totient number</a></div> A perfect totient number is an integer that is equal to the sum of its iterated totients. That is, we apply the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number. <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=21" title="Edit section: Applications">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Cyclotomy">Cyclotomy</h3>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=22" title="Edit section: Cyclotomy">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/Constructible_polygon" title="Constructible polygon">Constructible polygon</a></div> In the last section of the <a href="/wiki/Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones</a><a href="#cite_note-49">[49]</a><a href="#cite_note-50">[50]</a> Gauss proves<a href="#cite_note-51">[51]</a> that a regular n-gon can be constructed with straightedge and compass if φ(n) is a power of 2. If n is a power of an odd prime number the formula for the totient says its totient can be a power of two only if n is a first power and n − 1 is a power of 2. The primes that are one more than a power of 2 are called <a href="/wiki/Fermat_prime" class="mw-redirect" title="Fermat prime">Fermat primes</a>, and only five are known: 3, 5, 17, 257, and 65537. Fermat and Gauss knew of these. Nobody has been able to prove whether there are any more. Thus, a regular n-gon has a straightedge-and-compass construction if n is a product of distinct Fermat primes and any power of 2. The first few such n are<a href="#cite_note-52">[52]</a> <dl><dd>2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40,... (sequence <a href="//oeis.org/A003401" class="extiw" title="oeis:A003401">A003401</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Prime_number_theorem_for_arithmetic_progressions">Prime number theorem for arithmetic progressions</h3>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=23" title="Edit section: Prime number theorem for arithmetic progressions">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/Prime_number_theorem#Prime_number_theorem_for_arithmetic_progressions" title="Prime number theorem">Prime number theorem § Prime number theorem for arithmetic progressions</a></div> <div class="mw-heading mw-heading3"><h3 id="The_RSA_cryptosystem">The RSA cryptosystem</h3>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=24" title="Edit section: The RSA cryptosystem">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/RSA_(algorithm)" class="mw-redirect" title="RSA (algorithm)">RSA (algorithm)</a></div> Setting up an RSA system involves choosing large prime numbers p and q, computing n = pq and k = φ(n), and finding two numbers e and d such that ed ≡ 1 (mod k). The numbers n and e (the "encryption key") are released to the public, and d (the "decryption key") is kept private. A message, represented by an integer m, where 0 < m < n, is encrypted by computing S = me (mod n). It is decrypted by computing t = Sd (mod n). Euler's Theorem can be used to show that if 0 < t < n, then t = m. The security of an RSA system would be compromised if the number n could be efficiently factored or if φ(n) could be efficiently computed without factoring n. <div class="mw-heading mw-heading2"><h2 id="Unsolved_problems">Unsolved problems</h2>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=25" title="Edit section: Unsolved problems">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Lehmer's_conjecture">Lehmer's conjecture</h3>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=26" title="Edit section: Lehmer's conjecture">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/Lehmer%27s_totient_problem" title="Lehmer's totient problem">Lehmer's totient problem</a></div> If p is prime, then φ(p) = p − 1. In 1932 <a href="/wiki/D._H._Lehmer" title="D. H. Lehmer">D. H. Lehmer</a> asked if there are any composite numbers n such that φ(n) divides n − 1. None are known.<a href="#cite_note-53">[53]</a> In 1933 he proved that if any such n exists, it must be odd, square-free, and divisible by at least seven primes (i.e. ω(n) ≥ 7). In 1980 Cohen and Hagis proved that n > 1020 and that ω(n) ≥ 14.<a href="#cite_note-54">[54]</a> Further, Hagis showed that if 3 divides n then n > 101937042 and ω(n) ≥ 298848.<a href="#cite_note-55">[55]</a><a href="#cite_note-Guy142-56">[56]</a> <div class="mw-heading mw-heading3"><h3 id="Carmichael's_conjecture">Carmichael's conjecture</h3>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=27" title="Edit section: Carmichael's conjecture">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/Carmichael%27s_totient_function_conjecture" title="Carmichael's totient function conjecture">Carmichael's totient function conjecture</a></div> This states that there is no number n with the property that for all other numbers m, m ≠ n, φ(m) ≠ φ(n). See <a href="#Ford's_theorem">Ford's theorem</a> above. As stated in the main article, if there is a single counterexample to this conjecture, there must be infinitely many counterexamples, and the smallest one has at least ten billion digits in base 10.<a href="#cite_note-Guy144-40">[40]</a> <div class="mw-heading mw-heading3"><h3 id="Riemann_hypothesis">Riemann hypothesis</h3>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=28" title="Edit section: Riemann hypothesis">edit</a>]</div> The <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> is true if and only if the inequality <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n}{\varphi (n)}}<e^{\gamma }\log \log n+{\frac {e^{\gamma }(4+\gamma -\log 4\pi )}{\sqrt {\log n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo><</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>4</mn> <mo>+</mo> <mi>γ</mi> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mn>4</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n}{\varphi (n)}}<e^{\gamma }\log \log n+{\frac {e^{\gamma }(4+\gamma -\log 4\pi )}{\sqrt {\log n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fee6f34ce047c953cc5a99479e5c587642abdc12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.02ex; height:7.009ex;" alt="{\displaystyle {\frac {n}{\varphi (n)}}<e^{\gamma }\log \log n+{\frac {e^{\gamma }(4+\gamma -\log 4\pi )}{\sqrt {\log n}}}}"></dd></dl> is true for all n ≥ p120569# where γ is <a href="/wiki/Euler%27s_constant" title="Euler's constant">Euler's constant</a> and p120569# is the <a href="/wiki/Primorial" title="Primorial">product of the first</a> 120569 primes.<a href="#cite_note-57">[57]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=29" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Carmichael_function" title="Carmichael function">Carmichael function (λ)</a></li> <li><a href="/wiki/Dedekind_psi_function" title="Dedekind psi function">Dedekind psi function (𝜓)</a></li> <li><a href="/wiki/Divisor_function" title="Divisor function">Divisor function (σ)</a></li></ul> <ul><li><a href="/wiki/Duffin%E2%80%93Schaeffer_conjecture" class="mw-redirect" title="Duffin–Schaeffer conjecture">Duffin–Schaeffer conjecture</a></li> <li><a href="/wiki/Fermat%27s_little_theorem#Generalizations" title="Fermat's little theorem">Generalizations of Fermat's little theorem</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite number</a></li></ul> <ul><li><a href="/wiki/Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">Multiplicative group of integers modulo n</a></li> <li><a href="/wiki/Ramanujan_sum" class="mw-redirect" title="Ramanujan sum">Ramanujan sum</a></li> <li><a href="/wiki/Totient_summatory_function" title="Totient summatory function">Totient summatory function (𝛷)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=30" title="Edit section: Notes">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: 30em;"> <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 rel="nofollow" class="external text" href="https://www.khanacademy.org/computing/computer-science/cryptography/modern-crypt/v/euler-s-totient-function-phi-function">"Euler's totient function"</a>. Khan Academy. Retrieved 2016-02-26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Khan+Academy&rft.atitle=Euler%27s+totient+function&rft_id=https%3A%2F%2Fwww.khanacademy.org%2Fcomputing%2Fcomputer-science%2Fcryptography%2Fmodern-crypt%2Fv%2Feuler-s-totient-function-phi-function&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFLong1972">Long (1972</a>, p. 85) </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFPettofrezzoByrkit1970">Pettofrezzo & Byrkit (1970</a>, p. 72) </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFLong1972">Long (1972</a>, p. 162) </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFPettofrezzoByrkit1970">Pettofrezzo & Byrkit (1970</a>, p. 80) </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> See <a href="#Euler's_theorem">Euler's theorem</a>. </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> L. Euler "<a rel="nofollow" class="external text" href="http://eulerarchive.maa.org/pages/E271.html">Theoremata arithmetica nova methodo demonstrata</a>" (An arithmetic theorem proved by a new method), Novi commentarii academiae scientiarum imperialis Petropolitanae (New Memoirs of the Saint-Petersburg Imperial Academy of Sciences), 8 (1763), 74–104. (The work was presented at the Saint-Petersburg Academy on October 15, 1759. A work with the same title was presented at the Berlin Academy on June 8, 1758). Available on-line in: <a href="/wiki/Ferdinand_Rudio" title="Ferdinand Rudio">Ferdinand Rudio</a>, <abbr title="editor">ed.</abbr>, Leonhardi Euleri Commentationes Arithmeticae, volume 1, in: Leonhardi Euleri Opera Omnia, series 1, volume 2 (Leipzig, Germany, B. G. Teubner, 1915), <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k6952c/f571.image">pages 531–555</a>. On page 531, Euler defines n as the number of integers that are smaller than N and relatively prime to N (... aequalis sit multitudini numerorum ipso N minorum, qui simul ad eum sint primi, ...), which is the phi function, φ(N). </li> <li id="cite_note-Sandifer,_p._203-8">^ <a href="#cite_ref-Sandifer,_p._203_8-0">a</a> <a href="#cite_ref-Sandifer,_p._203_8-1">b</a> Sandifer, p. 203 </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> Graham et al. p. 133 note 111 </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> L. Euler, <a rel="nofollow" class="external text" href="http://math.dartmouth.edu/~euler/docs/originals/E564.pdf">Speculationes circa quasdam insignes proprietates numerorum</a>, Acta Academiae Scientarum Imperialis Petropolitinae, vol. 4, (1784), pp. 18–30, or Opera Omnia, Series 1, volume 4, pp. 105–115. (The work was presented at the Saint-Petersburg Academy on October 9, 1775). </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> Both φ(n) and ϕ(n) are seen in the literature. These are two forms of the lower-case Greek letter <a href="/wiki/Phi" title="Phi">phi</a>. </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Gauss, Disquisitiones Arithmeticae article 38 </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCajori1929" class="citation book cs1"><a href="/wiki/Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1929). A History Of Mathematical Notations Volume II. Open Court Publishing Company. §409.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+Of+Mathematical+Notations+Volume+II&rft.pages=%C2%A7409&rft.pub=Open+Court+Publishing+Company&rft.date=1929&rft.aulast=Cajori&rft.aufirst=Florian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> J. J. Sylvester (1879) "On certain ternary cubic-form equations", American Journal of Mathematics, 2 : 357-393; Sylvester coins the term "totient" on <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-AcPAAAAIAAJ&pg=PA361">page 361</a>. </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReference-OED2-totient" class="citation book cs1">"totient". <a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a> (2nd ed.). <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. 1989.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=totient&rft.btitle=Oxford+English+Dictionary&rft.edition=2nd&rft.pub=Oxford+University+Press&rft.date=1989&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <a href="#CITEREFSchramm2008">Schramm (2008)</a> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> Gauss, DA, art 39 </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> Gauss, DA art. 39, arts. 52-54 </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> Graham et al. pp. 134-135 </li> <li id="cite_note-hw328-20">^ <a href="#cite_ref-hw328_20-0">a</a> <a href="#cite_ref-hw328_20-1">b</a> <a href="#CITEREFHardyWright1979">Hardy & Wright 1979</a>, thm. 328 </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> Dineva (in external refs), prop. 1 </li> <li id="cite_note-Wal1963-22">^ <a href="#cite_ref-Wal1963_22-0">a</a> <a href="#cite_ref-Wal1963_22-1">b</a> <a href="#cite_ref-Wal1963_22-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWalfisz1963" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Arnold_Walfisz" title="Arnold Walfisz">Walfisz, Arnold</a> (1963). Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte (in German). Vol. 16. Berlin: <a href="/wiki/VEB_Deutscher_Verlag_der_Wissenschaften" class="mw-redirect" title="VEB Deutscher Verlag der Wissenschaften">VEB Deutscher Verlag der Wissenschaften</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0146.06003">0146.06003</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Weylsche+Exponentialsummen+in+der+neueren+Zahlentheorie&rft.place=Berlin&rft.series=Mathematische+Forschungsberichte&rft.pub=VEB+Deutscher+Verlag+der+Wissenschaften&rft.date=1963&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0146.06003%23id-name%3DZbl&rft.aulast=Walfisz&rft.aufirst=Arnold&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLomadse1964" class="citation cs2">Lomadse, G. (1964), <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/aa/aa10/aa10111.pdf">"The scientific work of Arnold Walfisz"</a> (PDF), Acta Arithmetica, 10 (3): 227–237, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Faa-10-3-227-237">10.4064/aa-10-3-227-237</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Arithmetica&rft.atitle=The+scientific+work+of+Arnold+Walfisz&rft.volume=10&rft.issue=3&rft.pages=227-237&rft.date=1964&rft_id=info%3Adoi%2F10.4064%2Faa-10-3-227-237&rft.aulast=Lomadse&rft.aufirst=G.&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fksiazki%2Faa%2Faa10%2Faa10111.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"> </li> <li id="cite_note-Sita-24">^ <a href="#cite_ref-Sita_24-0">a</a> <a href="#cite_ref-Sita_24-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSitaramachandrarao1985" class="citation journal cs1">Sitaramachandrarao, R. (1985). <a rel="nofollow" class="external text" href="https://doi.org/10.1216%2FRMJ-1985-15-2-579">"On an error term of Landau II"</a>. Rocky Mountain J. Math. 15 (2): 579–588. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1216%2FRMJ-1985-15-2-579">10.1216/RMJ-1985-15-2-579</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rocky+Mountain+J.+Math.&rft.atitle=On+an+error+term+of+Landau+II&rft.volume=15&rft.issue=2&rft.pages=579-588&rft.date=1985&rft_id=info%3Adoi%2F10.1216%2FRMJ-1985-15-2-579&rft.aulast=Sitaramachandrarao&rft.aufirst=R.&rft_id=https%3A%2F%2Fdoi.org%2F10.1216%252FRMJ-1985-15-2-579&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"> </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPollack2023" class="citation cs2">Pollack, P. (2023), "Two problems on the distribution of Carmichael's lambda function", Mathematika, 69: 1195–1220, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2303.14043">2303.14043</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fmtk.12222">10.1112/mtk.12222</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematika&rft.atitle=Two+problems+on+the+distribution+of+Carmichael%27s+lambda+function&rft.volume=69&rft.pages=1195-1220&rft.date=2023&rft_id=info%3Aarxiv%2F2303.14043&rft_id=info%3Adoi%2F10.1112%2Fmtk.12222&rft.aulast=Pollack&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="#CITEREFHardyWright1979">Hardy & Wright 1979</a>, thm. 288 </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <a href="#CITEREFHardyWright1979">Hardy & Wright 1979</a>, thm. 309 </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <a href="#CITEREFHardyWright1979">Hardy & Wright 1979</a>, intro to § 18.4 </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <a href="#CITEREFHardyWright1979">Hardy & Wright 1979</a>, thm. 326 </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <a href="#CITEREFHardyWright1979">Hardy & Wright 1979</a>, thm. 327 </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> In fact Chebyshev's theorem (<a href="#CITEREFHardyWright1979">Hardy & Wright 1979</a>, thm.7) and Mertens' third theorem is all that is needed. </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <a href="#CITEREFHardyWright1979">Hardy & Wright 1979</a>, thm. 436 </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> Theorem 15 of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosserSchoenfeld1962" class="citation journal cs1">Rosser, J. Barkley; Schoenfeld, Lowell (1962). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.ijm/1255631807">"Approximate formulas for some functions of prime numbers"</a>. Illinois J. Math. 6 (1): 64–94. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fijm%2F1255631807">10.1215/ijm/1255631807</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Illinois+J.+Math.&rft.atitle=Approximate+formulas+for+some+functions+of+prime+numbers&rft.volume=6&rft.issue=1&rft.pages=64-94&rft.date=1962&rft_id=info%3Adoi%2F10.1215%2Fijm%2F1255631807&rft.aulast=Rosser&rft.aufirst=J.+Barkley&rft.au=Schoenfeld%2C+Lowell&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.ijm%2F1255631807&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> Bach & Shallit, thm. 8.8.7 </li> <li id="cite_note-Rib320-35">^ <a href="#cite_ref-Rib320_35-0">a</a> <a href="#cite_ref-Rib320_35-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRibenboim1989" class="citation book cs1">Ribenboim (1989). "How are the Prime Numbers Distributed? §I.C The Distribution of Values of Euler's Function". The Book of Prime Number Records (2nd ed.). New York: Springer-Verlag. pp. 172–175. <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%2F978-1-4684-0507-1_5">10.1007/978-1-4684-0507-1_5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4684-0509-5" title="Special:BookSources/978-1-4684-0509-5"><bdi>978-1-4684-0509-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=How+are+the+Prime+Numbers+Distributed%3F+%C2%A7I.C+The+Distribution+of+Values+of+Euler%27s+Function&rft.btitle=The+Book+of+Prime+Number+Records&rft.place=New+York&rft.pages=172-175&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1989&rft_id=info%3Adoi%2F10.1007%2F978-1-4684-0507-1_5&rft.isbn=978-1-4684-0509-5&rft.au=Ribenboim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"> </li> <li id="cite_note-SMC2425-36"><a href="#cite_ref-SMC2425_36-0">^</a> Sándor, Mitrinović & Crstici (2006) pp.24–25 </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <a href="#CITEREFHardyWright1979">Hardy & Wright 1979</a>, thm. 332 </li> <li id="cite_note-Rib38-38">^ <a href="#cite_ref-Rib38_38-0">a</a> <a href="#cite_ref-Rib38_38-1">b</a> <a href="#cite_ref-Rib38_38-2">c</a> Ribenboim, p.38 </li> <li id="cite_note-SMC16-39">^ <a href="#cite_ref-SMC16_39-0">a</a> <a href="#cite_ref-SMC16_39-1">b</a> Sándor, Mitrinović & Crstici (2006) p.16 </li> <li id="cite_note-Guy144-40">^ <a href="#cite_ref-Guy144_40-0">a</a> <a href="#cite_ref-Guy144_40-1">b</a> Guy (2004) p.144 </li> <li id="cite_note-SC230-41"><a href="#cite_ref-SC230_41-0">^</a> Sándor & Crstici (2004) p.230 </li> <li id="cite_note-Zha1993-42"><a href="#cite_ref-Zha1993_42-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZhang1993" class="citation journal cs1">Zhang, Mingzhi (1993). <a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjnth.1993.1014">"On nontotients"</a>. <a href="/wiki/Journal_of_Number_Theory" title="Journal of Number Theory">Journal of Number Theory</a>. 43 (2): 168–172. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjnth.1993.1014">10.1006/jnth.1993.1014</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-314X">0022-314X</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0772.11001">0772.11001</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=On+nontotients&rft.volume=43&rft.issue=2&rft.pages=168-172&rft.date=1993&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0772.11001%23id-name%3DZbl&rft.issn=0022-314X&rft_id=info%3Adoi%2F10.1006%2Fjnth.1993.1014&rft.aulast=Zhang&rft.aufirst=Mingzhi&rft_id=https%3A%2F%2Fdoi.org%2F10.1006%252Fjnth.1993.1014&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"> </li> <li id="cite_note-Ford1998-43">^ <a href="#cite_ref-Ford1998_43-0">a</a> <a href="#cite_ref-Ford1998_43-1">b</a> <a href="#cite_ref-Ford1998_43-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFord1998" class="citation journal cs1">Ford, Kevin (1998). "The distribution of totients". Ramanujan J. Developments in Mathematics. 2 (1–2): 67–151. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1104.3264">1104.3264</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4757-4507-8_8">10.1007/978-1-4757-4507-8_8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-5058-1" title="Special:BookSources/978-1-4419-5058-1"><bdi>978-1-4419-5058-1</bdi></a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1382-4090">1382-4090</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0914.11053">0914.11053</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ramanujan+J.&rft.atitle=The+distribution+of+totients&rft.volume=2&rft.issue=1%E2%80%932&rft.pages=67-151&rft.date=1998&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0914.11053%23id-name%3DZbl&rft_id=info%3Adoi%2F10.1007%2F978-1-4757-4507-8_8&rft_id=info%3Aarxiv%2F1104.3264&rft.issn=1382-4090&rft.isbn=978-1-4419-5058-1&rft.aulast=Ford&rft.aufirst=Kevin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"> </li> <li id="cite_note-SMC22-44"><a href="#cite_ref-SMC22_44-0">^</a> Sándor et al (2006) p.22 </li> <li id="cite_note-SMC21-45"><a href="#cite_ref-SMC21_45-0">^</a> Sándor et al (2006) p.21 </li> <li id="cite_note-Guy145-46">^ <a href="#cite_ref-Guy145_46-0">a</a> <a href="#cite_ref-Guy145_46-1">b</a> Guy (2004) p.145 </li> <li id="cite_note-SC229-47"><a href="#cite_ref-SC229_47-0">^</a> Sándor & Crstici (2004) p.229 </li> <li id="cite_note-SC228-48"><a href="#cite_ref-SC228_48-0">^</a> Sándor & Crstici (2004) p.228 </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> Gauss, DA. The 7th § is arts. 336–366 </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> Gauss proved if n satisfies certain conditions then the n-gon can be constructed. In 1837 <a href="/wiki/Pierre_Wantzel" title="Pierre Wantzel">Pierre Wantzel</a> proved the converse, if the n-gon is constructible, then n must satisfy Gauss's conditions </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> Gauss, DA, art 366 </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> Gauss, DA, art. 366. This list is the last sentence in the Disquisitiones </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> Ribenboim, pp. 36–37. </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="CITEREFCohenHagis1980" class="citation journal cs1">Cohen, Graeme L.; Hagis, Peter Jr. (1980). "On the number of prime factors of n if φ(n) divides n − 1". Nieuw Arch. Wiskd. III Series. 28: 177–185. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0028-9825">0028-9825</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0436.10002">0436.10002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nieuw+Arch.+Wiskd.&rft.atitle=On+the+number+of+prime+factors+of+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3En%3C%2Fspan%3E+if+%3Cspan+class%3D%22texhtml+%22+%3E%CF%86%28n%29%3C%2Fspan%3E+divides+%3Cspan+class%3D%22texhtml+%22+%3En+%E2%88%92+1%3C%2Fspan%3E&rft.volume=28&rft.pages=177-185&rft.date=1980&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0436.10002%23id-name%3DZbl&rft.issn=0028-9825&rft.aulast=Cohen&rft.aufirst=Graeme+L.&rft.au=Hagis%2C+Peter+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+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 id="CITEREFHagis1988" class="citation journal cs1">Hagis, Peter Jr. (1988). "On the equation M·φ(n) = n − 1". Nieuw Arch. Wiskd. IV Series. 6 (3): 255–261. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0028-9825">0028-9825</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0668.10006">0668.10006</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nieuw+Arch.+Wiskd.&rft.atitle=On+the+equation+%3Cspan+class%3D%22texhtml+%22+%3EM%C2%B7%CF%86%28n%29+%3D+n+%E2%88%92+1%3C%2Fspan%3E&rft.volume=6&rft.issue=3&rft.pages=255-261&rft.date=1988&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0668.10006%23id-name%3DZbl&rft.issn=0028-9825&rft.aulast=Hagis&rft.aufirst=Peter+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"> </li> <li id="cite_note-Guy142-56"><a href="#cite_ref-Guy142_56-0">^</a> Guy (2004) p.142 </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="CITEREFBroughan2017" class="citation book cs1">Broughan, Kevin (2017). Equivalents of the Riemann Hypothesis, Volume One: Arithmetic Equivalents (First ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-19704-6" title="Special:BookSources/978-1-107-19704-6"><bdi>978-1-107-19704-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Equivalents+of+the+Riemann+Hypothesis%2C+Volume+One%3A+Arithmetic+Equivalents&rft.edition=First&rft.pub=Cambridge+University+Press&rft.date=2017&rft.isbn=978-1-107-19704-6&rft.aulast=Broughan&rft.aufirst=Kevin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"> Corollary 5.35 </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Euler%27s_totient_function&action=edit&section=31" title="Edit section: References">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: 30em"> The <a href="/wiki/Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a> has been translated from Latin into English and German. The German edition includes all of Gauss's papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. References to the Disquisitiones are of the form Gauss, DA, art. nnn. <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbramowitzStegun1964" class="citation cs2"><a href="/wiki/Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, M.</a>; <a href="/wiki/Irene_A._Stegun" class="mw-redirect" title="Irene A. Stegun">Stegun, I. A.</a> (1964), <a rel="nofollow" class="external text" href="https://archive.org/details/handbookofmathe000abra">Handbook of Mathematical Functions</a>, New York: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-61272-4" title="Special:BookSources/0-486-61272-4"><bdi>0-486-61272-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Mathematical+Functions&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1964&rft.isbn=0-486-61272-4&rft.aulast=Abramowitz&rft.aufirst=M.&rft.au=Stegun%2C+I.+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhandbookofmathe000abra&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988">. See paragraph 24.3.2.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBachShallit1996" class="citation cs2"><a href="/wiki/Eric_Bach" title="Eric Bach">Bach, Eric</a>; <a href="/wiki/Jeffrey_Shallit" title="Jeffrey Shallit">Shallit, Jeffrey</a> (1996), Algorithmic Number Theory (Vol I: Efficient Algorithms), MIT Press Series in the Foundations of Computing, Cambridge, MA: <a href="/wiki/The_MIT_Press" class="mw-redirect" title="The MIT Press">The MIT Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-02405-5" title="Special:BookSources/0-262-02405-5"><bdi>0-262-02405-5</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0873.11070">0873.11070</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithmic+Number+Theory+%28Vol+I%3A+Efficient+Algorithms%29&rft.place=Cambridge%2C+MA&rft.series=MIT+Press+Series+in+the+Foundations+of+Computing&rft.pub=The+MIT+Press&rft.date=1996&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0873.11070%23id-name%3DZbl&rft.isbn=0-262-02405-5&rft.aulast=Bach&rft.aufirst=Eric&rft.au=Shallit%2C+Jeffrey&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"></li> <li>Dickson, Leonard Eugene, "History Of The Theory Of Numbers", vol 1, chapter 5 "Euler's Function, Generalizations; Farey Series", Chelsea Publishing 1952</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFord1999" class="citation cs2">Ford, Kevin (1999), "The number of solutions of φ(x) = m", <a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>, 150 (1): 283–311, <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%2F121103">10.2307/121103</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-486X">0003-486X</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/121103">121103</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=1715326">1715326</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0978.11053">0978.11053</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=The+number+of+solutions+of+%CF%86%28x%29+%3D+m&rft.volume=150&rft.issue=1&rft.pages=283-311&rft.date=1999&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0978.11053%23id-name%3DZbl&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F121103%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F121103&rft.issn=0003-486X&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1715326%23id-name%3DMR&rft.aulast=Ford&rft.aufirst=Kevin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGauss1986" class="citation cs2"><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss, Carl Friedrich</a> (1986), Disquisitiones Arithmeticae (Second, corrected edition), translated by Clarke, Arthur A., New York: <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-96254-9" title="Special:BookSources/0-387-96254-9"><bdi>0-387-96254-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Disquisitiones+Arithmeticae+%28Second%2C+corrected+edition%29&rft.place=New+York&rft.pub=Springer&rft.date=1986&rft.isbn=0-387-96254-9&rft.aulast=Gauss&rft.aufirst=Carl+Friedrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGauss1965" class="citation cs2"><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss, Carl Friedrich</a> (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition), translated by Maser, H., New York: Chelsea, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8284-0191-8" title="Special:BookSources/0-8284-0191-8"><bdi>0-8284-0191-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Untersuchungen+uber+hohere+Arithmetik+%28Disquisitiones+Arithmeticae+%26+other+papers+on+number+theory%29+%28Second+edition%29&rft.place=New+York&rft.pub=Chelsea&rft.date=1965&rft.isbn=0-8284-0191-8&rft.aulast=Gauss&rft.aufirst=Carl+Friedrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrahamKnuthPatashnik1994" class="citation cs2"><a href="/wiki/Ronald_Graham" title="Ronald Graham">Graham, Ronald</a>; <a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a>; <a href="/wiki/Oren_Patashnik" title="Oren Patashnik">Patashnik, Oren</a> (1994), <a href="/wiki/Concrete_Mathematics" title="Concrete Mathematics">Concrete Mathematics</a>: a foundation for computer science (2nd ed.), Reading, MA: Addison-Wesley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-55802-5" title="Special:BookSources/0-201-55802-5"><bdi>0-201-55802-5</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0836.00001">0836.00001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Concrete+Mathematics%3A+a+foundation+for+computer+science&rft.place=Reading%2C+MA&rft.edition=2nd&rft.pub=Addison-Wesley&rft.date=1994&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0836.00001%23id-name%3DZbl&rft.isbn=0-201-55802-5&rft.aulast=Graham&rft.aufirst=Ronald&rft.au=Knuth%2C+Donald&rft.au=Patashnik%2C+Oren&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuy2004" class="citation cs2"><a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, Richard K.</a> (2004), Unsolved Problems in Number Theory, Problem Books in Mathematics (3rd ed.), New York, NY: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-20860-7" title="Special:BookSources/0-387-20860-7"><bdi>0-387-20860-7</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1058.11001">1058.11001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unsolved+Problems+in+Number+Theory&rft.place=New+York%2C+NY&rft.series=Problem+Books+in+Mathematics&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=2004&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1058.11001%23id-name%3DZbl&rft.isbn=0-387-20860-7&rft.aulast=Guy&rft.aufirst=Richard+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardyWright1979" class="citation cs2"><a href="/wiki/G._H._Hardy" title="G. H. Hardy">Hardy, G. H.</a>; <a href="/wiki/E._M._Wright" title="E. M. Wright">Wright, E. M.</a> (1979), <a href="/wiki/An_Introduction_to_the_Theory_of_Numbers" title="An Introduction to the Theory of Numbers">An Introduction to the Theory of Numbers</a> (Fifth ed.), Oxford: <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853171-5" title="Special:BookSources/978-0-19-853171-5"><bdi>978-0-19-853171-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Theory+of+Numbers&rft.place=Oxford&rft.edition=Fifth&rft.pub=Oxford+University+Press&rft.date=1979&rft.isbn=978-0-19-853171-5&rft.aulast=Hardy&rft.aufirst=G.+H.&rft.au=Wright%2C+E.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiu2016" class="citation cs2">Liu, H.-Q. (2016), "On Euler's function", Proc. Roy. Soc. Edinburgh Sect. A, 146 (4)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+Roy.+Soc.+Edinburgh+Sect.+A&rft.atitle=On+Euler%27s+function&rft.volume=146&rft.issue=4&rft.date=2016&rft.aulast=Liu&rft.aufirst=H.-Q.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLong1972" class="citation cs2">Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: <a href="/wiki/D._C._Heath_and_Company" title="D. C. Heath and Company">D. C. Heath and Company</a>, <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/77-171950">77-171950</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Introduction+to+Number+Theory&rft.place=Lexington&rft.edition=2nd&rft.pub=D.+C.+Heath+and+Company&rft.date=1972&rft_id=info%3Alccn%2F77-171950&rft.aulast=Long&rft.aufirst=Calvin+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPettofrezzoByrkit1970" class="citation cs2">Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>, <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/77-81766">77-81766</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Number+Theory&rft.place=Englewood+Cliffs&rft.pub=Prentice+Hall&rft.date=1970&rft_id=info%3Alccn%2F77-81766&rft.aulast=Pettofrezzo&rft.aufirst=Anthony+J.&rft.au=Byrkit%2C+Donald+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRibenboim1996" class="citation cs2"><a href="/wiki/Paulo_Ribenboim" title="Paulo Ribenboim">Ribenboim, Paulo</a> (1996), The New Book of Prime Number Records (3rd ed.), New York: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94457-5" title="Special:BookSources/0-387-94457-5"><bdi>0-387-94457-5</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0856.11001">0856.11001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+New+Book+of+Prime+Number+Records&rft.place=New+York&rft.edition=3rd&rft.pub=Springer&rft.date=1996&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0856.11001%23id-name%3DZbl&rft.isbn=0-387-94457-5&rft.aulast=Ribenboim&rft.aufirst=Paulo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSandifer2007" class="citation cs2">Sandifer, Charles (2007), The early mathematics of Leonhard Euler, MAA, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-559-1" title="Special:BookSources/978-0-88385-559-1"><bdi>978-0-88385-559-1</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+early+mathematics+of+Leonhard+Euler&rft.pub=MAA&rft.date=2007&rft.isbn=978-0-88385-559-1&rft.aulast=Sandifer&rft.aufirst=Charles&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSándorMitrinovićCrstici2006" class="citation cs2">Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006), Handbook of number theory I, Dordrecht: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, pp. 9–36, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4020-4215-9" title="Special:BookSources/1-4020-4215-9"><bdi>1-4020-4215-9</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1151.11300">1151.11300</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+number+theory+I&rft.place=Dordrecht&rft.pages=9-36&rft.pub=Springer-Verlag&rft.date=2006&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1151.11300%23id-name%3DZbl&rft.isbn=1-4020-4215-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSándorCrstici2004" class="citation book cs1">Sándor, Jozsef; Crstici, Borislav (2004). <a rel="nofollow" class="external text" href="https://archive.org/details/handbooknumberth00sand_741">Handbook of number theory II</a>. Dordrecht: Kluwer Academic. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/handbooknumberth00sand_741/page/n179">179</a>–327. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4020-2546-7" title="Special:BookSources/1-4020-2546-7"><bdi>1-4020-2546-7</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1079.11001">1079.11001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+number+theory+II&rft.place=Dordrecht&rft.pages=179-327&rft.pub=Kluwer+Academic&rft.date=2004&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1079.11001%23id-name%3DZbl&rft.isbn=1-4020-2546-7&rft.aulast=S%C3%A1ndor&rft.aufirst=Jozsef&rft.au=Crstici%2C+Borislav&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhandbooknumberth00sand_741&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchramm2008" class="citation cs2">Schramm, Wolfgang (2008), <a rel="nofollow" class="external text" href="http://www.integers-ejcnt.org/vol8.html">"The Fourier transform of functions of the greatest common divisor"</a>, Electronic Journal of Combinatorial Number Theory, A50 (8(1))</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Electronic+Journal+of+Combinatorial+Number+Theory&rft.atitle=The+Fourier+transform+of+functions+of+the+greatest+common+divisor&rft.volume=A50&rft.issue=8%281%29&rft.date=2008&rft.aulast=Schramm&rft.aufirst=Wolfgang&rft_id=http%3A%2F%2Fwww.integers-ejcnt.org%2Fvol8.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+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=Euler%27s_totient_function&action=edit&section=32" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Totient_function">"Totient 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=Totient+function&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DTotient_function&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+totient+function" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://mathcenter.oxford.emory.edu/site/math125/chineseRemainderTheorem/">Euler's Phi Function and the Chinese Remainder Theorem — proof that φ(n) is multiplicative</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210228071226/http://mathcenter.oxford.emory.edu/site/math125/chineseRemainderTheorem/">Archived</a> 2021-02-28 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.javascripter.net/math/calculators/eulertotientfunction.htm">Euler's totient function calculator in JavaScript — up to 20 digits</a></li> <li>Dineva, Rosica, <a rel="nofollow" class="external text" href="http://www.mtholyoke.edu/~robinson/reu/reu05/rdineva1.pdf">The Euler Totient, the Möbius, and the Divisor Functions</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210116061553/https://www.mtholyoke.edu/~robinson/reu/reu05/rdineva1.pdf">Archived</a> 2021-01-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li>Plytage, Loomis, Polhill <a rel="nofollow" class="external text" href="http://facstaff.bloomu.edu/jpolhill/cmj034-042.pdf">Summing Up The Euler Phi Function</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output 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.navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Totient" title="Template:Totient"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Totient" title="Template talk:Totient"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Totient" title="Special:EditPage/Template:Totient"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Totient_function" style="font-size:114%;margin:0 4em">Totient function</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Euler's totient function</a> φ(n)</li> <li><a href="/wiki/Jordan%27s_totient_function" title="Jordan's totient function">Jordan's totient function</a> Jk(n)</li> <li><a href="/wiki/Carmichael_function" title="Carmichael function">Carmichael function</a> (reduced totient function) λ(n)</li> <li><a href="/wiki/Nontotient" title="Nontotient">Nontotient</a></li> <li><a href="/wiki/Noncototient" title="Noncototient">Noncototient</a></li> <li><a href="/wiki/Highly_totient_number" title="Highly totient number">Highly totient number</a></li> <li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient number</a></li> <li><a href="/wiki/Sparsely_totient_number" title="Sparsely totient number">Sparsely totient number</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" 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Euler's totient function - Wikipedia