Arithmetic function

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class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Notation</span> </div> </a> <ul id="toc-Notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ω(n),_ω(n),_νp(n)_–_prime_power_decomposition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ω(n),_ω(n),_νp(n)_–_prime_power_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ω(<i>n</i>), <i>ω</i>(<i>n</i>), <i>ν</i><sub><i>p</i></sub>(<i>n</i>) – prime power decomposition</span> </div> </a> <ul id="toc-Ω(n),_ω(n),_νp(n)_–_prime_power_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplicative_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Multiplicative_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Multiplicative functions</span> </div> </a> <button aria-controls="toc-Multiplicative_functions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Multiplicative functions subsection</span> </button> <ul id="toc-Multiplicative_functions-sublist" class="vector-toc-list"> <li id="toc-σk(n),_τ(n),_d(n)_–_divisor_sums" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#σk(n),_τ(n),_d(n)_–_divisor_sums"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>σ<sub><i>k</i></sub>(<i>n</i>), τ(<i>n</i>), <i>d</i>(<i>n</i>) – divisor sums</span> </div> </a> <ul id="toc-σk(n),_τ(n),_d(n)_–_divisor_sums-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-φ(n)_–_Euler_totient_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#φ(n)_–_Euler_totient_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>φ(<i>n</i>) – Euler totient function</span> </div> </a> <ul id="toc-φ(n)_–_Euler_totient_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jk(n)_–_Jordan_totient_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Jk(n)_–_Jordan_totient_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>J<sub><i>k</i></sub>(<i>n</i>) – Jordan totient function</span> </div> </a> <ul id="toc-Jk(n)_–_Jordan_totient_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-μ(n)_–_Möbius_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#μ(n)_–_Möbius_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>μ(<i>n</i>) – Möbius function</span> </div> </a> <ul id="toc-μ(n)_–_Möbius_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-τ(n)_–_Ramanujan_tau_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#τ(n)_–_Ramanujan_tau_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>τ(<i>n</i>) – Ramanujan tau function</span> </div> </a> <ul id="toc-τ(n)_–_Ramanujan_tau_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-cq(n)_–_Ramanujan's_sum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#cq(n)_–_Ramanujan's_sum"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span><i>c</i><sub><i>q</i></sub>(<i>n</i>) – Ramanujan's sum</span> </div> </a> <ul id="toc-cq(n)_–_Ramanujan's_sum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ψ(n)_-_Dedekind_psi_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#ψ(n)_-_Dedekind_psi_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span><i>ψ</i>(<i>n</i>) - Dedekind psi function</span> </div> </a> <ul id="toc-ψ(n)_-_Dedekind_psi_function-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Completely_multiplicative_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Completely_multiplicative_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Completely multiplicative functions</span> </div> </a> <button aria-controls="toc-Completely_multiplicative_functions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Completely multiplicative functions subsection</span> </button> <ul id="toc-Completely_multiplicative_functions-sublist" class="vector-toc-list"> <li id="toc-λ(n)_–_Liouville_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#λ(n)_–_Liouville_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>λ(<i>n</i>) – Liouville function</span> </div> </a> <ul id="toc-λ(n)_–_Liouville_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-χ(n)_–_characters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#χ(n)_–_characters"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span><i>χ</i>(<i>n</i>) – characters</span> </div> </a> <ul id="toc-χ(n)_–_characters-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Additive_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Additive_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Additive functions</span> </div> </a> <button aria-controls="toc-Additive_functions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Additive functions subsection</span> </button> <ul id="toc-Additive_functions-sublist" class="vector-toc-list"> <li id="toc-ω(n)_–_distinct_prime_divisors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#ω(n)_–_distinct_prime_divisors"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span><i>ω</i>(<i>n</i>) – distinct prime divisors</span> </div> </a> <ul id="toc-ω(n)_–_distinct_prime_divisors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Completely_additive_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Completely_additive_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Completely additive functions</span> </div> </a> <button aria-controls="toc-Completely_additive_functions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Completely additive functions subsection</span> </button> <ul id="toc-Completely_additive_functions-sublist" class="vector-toc-list"> <li id="toc-Ω(n)_–_prime_divisors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ω(n)_–_prime_divisors"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Ω(<i>n</i>) – prime divisors</span> </div> </a> <ul id="toc-Ω(n)_–_prime_divisors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-νp(n)_–_p-adic_valuation_of_an_integer_n" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#νp(n)_–_p-adic_valuation_of_an_integer_n"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span><i>ν</i><sub><i>p</i></sub>(<i>n</i>) – <i>p</i>-adic valuation of an integer <i>n</i></span> </div> </a> <ul id="toc-νp(n)_–_p-adic_valuation_of_an_integer_n-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logarithmic_derivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logarithmic_derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Logarithmic derivative</span> </div> </a> <ul id="toc-Logarithmic_derivative-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Neither_multiplicative_nor_additive" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Neither_multiplicative_nor_additive"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Neither multiplicative nor additive</span> </div> </a> <button aria-controls="toc-Neither_multiplicative_nor_additive-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Neither multiplicative nor additive subsection</span> </button> <ul id="toc-Neither_multiplicative_nor_additive-sublist" class="vector-toc-list"> <li id="toc-π(x),_Π(x),_θ(x),_ψ(x)_–_prime-counting_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#π(x),_Π(x),_θ(x),_ψ(x)_–_prime-counting_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span><span>π</span>(<i>x</i>), Π(<i>x</i>), <i>θ</i>(<i>x</i>), <i>ψ</i>(<i>x</i>) – prime-counting functions</span> </div> </a> <ul id="toc-π(x),_Π(x),_θ(x),_ψ(x)_–_prime-counting_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Λ(n)_–_von_Mangoldt_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Λ(n)_–_von_Mangoldt_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Λ(<i>n</i>) – von Mangoldt function</span> </div> </a> <ul id="toc-Λ(n)_–_von_Mangoldt_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-p(n)_–_partition_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#p(n)_–_partition_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span><i>p</i>(<i>n</i>) – partition function</span> </div> </a> <ul id="toc-p(n)_–_partition_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-λ(n)_–_Carmichael_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#λ(n)_–_Carmichael_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.4</span> <span>λ(<i>n</i>) – Carmichael function</span> </div> </a> <ul id="toc-λ(n)_–_Carmichael_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-h(n)_–_Class_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#h(n)_–_Class_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.5</span> <span><i>h</i>(<i>n</i>) – Class number</span> </div> </a> <ul id="toc-h(n)_–_Class_number-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-rk(n)_–_Sum_of_k_squares" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#rk(n)_–_Sum_of_k_squares"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.6</span> <span><i>r</i><sub><i>k</i></sub>(<i>n</i>) – Sum of <i>k</i> squares</span> </div> </a> <ul id="toc-rk(n)_–_Sum_of_k_squares-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-D(n)_–_Arithmetic_derivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#D(n)_–_Arithmetic_derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.7</span> <span><i>D</i>(<i>n</i>) – Arithmetic derivative</span> </div> </a> <ul id="toc-D(n)_–_Arithmetic_derivative-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Summation_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Summation_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Summation functions</span> </div> </a> <ul id="toc-Summation_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dirichlet_convolution" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dirichlet_convolution"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Dirichlet convolution</span> </div> </a> <ul id="toc-Dirichlet_convolution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relations_among_the_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relations_among_the_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Relations among the functions</span> </div> </a> <button aria-controls="toc-Relations_among_the_functions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Relations among the functions subsection</span> </button> <ul id="toc-Relations_among_the_functions-sublist" class="vector-toc-list"> <li id="toc-Dirichlet_convolutions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dirichlet_convolutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Dirichlet convolutions</span> </div> </a> <ul id="toc-Dirichlet_convolutions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sums_of_squares" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sums_of_squares"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.2</span> <span>Sums of squares</span> </div> </a> <ul id="toc-Sums_of_squares-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Divisor_sum_convolutions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Divisor_sum_convolutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.3</span> <span>Divisor sum convolutions</span> </div> </a> <ul id="toc-Divisor_sum_convolutions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Class_number_related" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Class_number_related"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.4</span> <span>Class number related</span> </div> </a> <ul id="toc-Class_number_related-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prime-count_related" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prime-count_related"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.5</span> <span>Prime-count related</span> </div> </a> <ul id="toc-Prime-count_related-sublist" class="vector-toc-list"> </ul> </li> <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"> <span class="vector-toc-numb">11.6</span> <span>Menon's identity</span> </div> </a> <ul id="toc-Menon's_identity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Miscellaneous" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Miscellaneous"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.7</span> <span>Miscellaneous</span> </div> </a> <ul id="toc-Miscellaneous-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-First_100_values_of_some_arithmetic_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#First_100_values_of_some_arithmetic_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>First 100 values of some arithmetic functions</span> </div> </a> <ul id="toc-First_100_values_of_some_arithmetic_functions-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"> <span class="vector-toc-numb">13</span> <span>Notes</span> </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"> <span class="vector-toc-numb">14</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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"> <span class="vector-toc-numb">16</span> <span>External links</span> </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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span 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Available in 29 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-29" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">29 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D9%84%D8%A9_%D8%AD%D8%B3%D8%A7%D8%A8%D9%8A%D8%A9" title="دالة حسابية – Arabic" lang="ar" hreflang="ar" data-title="دالة حسابية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_aritm%C3%A8tica" title="Funció aritmètica – Catalan" lang="ca" hreflang="ca" data-title="Funció aritmètica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Aritmetick%C3%A1_funkce" title="Aritmetická funkce – Czech" lang="cs" hreflang="cs" data-title="Aritmetická funkce" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zahlentheoretische_Funktion" title="Zahlentheoretische Funktion – German" lang="de" hreflang="de" data-title="Zahlentheoretische Funktion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_aritm%C3%A9tica" title="Función aritmética – Spanish" lang="es" hreflang="es" data-title="Función aritmética" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Aritmetika_funkcio" title="Aritmetika funkcio – Esperanto" lang="eo" hreflang="eo" data-title="Aritmetika funkcio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_%D8%AD%D8%B3%D8%A7%D8%A8%DB%8C" title="تابع حسابی – Persian" lang="fa" hreflang="fa" data-title="تابع حسابی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fonction_arithm%C3%A9tique" title="Fonction arithmétique – French" lang="fr" hreflang="fr" data-title="Fonction arithmétique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n_aritm%C3%A9tica" title="Función aritmética – Galician" lang="gl" hreflang="gl" data-title="Función aritmética" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%EB%A1%A0%EC%A0%81_%ED%95%A8%EC%88%98" title="수론적 함수 – Korean" lang="ko" hreflang="ko" data-title="수론적 함수" 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functie – Dutch" lang="nl" hreflang="nl" data-title="Rekenkundige functie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0%E8%AB%96%E7%9A%84%E9%96%A2%E6%95%B0" title="数論的関数 – Japanese" lang="ja" hreflang="ja" data-title="数論的関数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja_arytmetyczna" title="Funkcja arytmetyczna – Polish" lang="pl" hreflang="pl" data-title="Funkcja arytmetyczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_aritm%C3%A9tica" title="Função aritmética – Portuguese" lang="pt" hreflang="pt" data-title="Função aritmética" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Арифметическая функция – Russian" lang="ru" hreflang="ru" data-title="Арифметическая функция" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Aritmetick%C3%A1_funkcia" title="Aritmetická funkcia – Slovak" lang="sk" hreflang="sk" data-title="Aritmetická funkcia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Aritmeti%C4%8Dna_funkcija" title="Aritmetična funkcija – Slovenian" lang="sl" hreflang="sl" data-title="Aritmetična funkcija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Aritmeettinen_funktio" title="Aritmeettinen funktio – Finnish" lang="fi" hreflang="fi" data-title="Aritmeettinen funktio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Aritmetisk_funktion" title="Aritmetisk funktion – Swedish" 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.ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Prose plainlinks metadata ambox ambox-style ambox-Prose" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>is in <a href="/wiki/MOS:LIST" class="mw-redirect" title="MOS:LIST">list</a> format but may read better as <a href="/wiki/MOS:PROSE" class="mw-redirect" title="MOS:PROSE">prose</a></b>.<span class="hide-when-compact"> You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Arithmetic_function&action=edit">converting this article</a>, if appropriate. <a href="/wiki/Help:Editing" title="Help:Editing">Editing help</a> is available.</span> <span class="date-container"><i>(<span class="date">July 2020</span>)</i></span></div></td></tr></tbody></table> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Function whose domain is the positive integers</div> <p>In <a href="/wiki/Number_theory" title="Number theory">number theory</a>, an <b>arithmetic</b>, <b>arithmetical</b>, or <b>number-theoretic function</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> is generally any <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <i>f</i>(<i>n</i>) whose domain is the <a href="/wiki/Natural_number" title="Natural number">positive integers</a> and whose range is a <a href="/wiki/Subset" title="Subset">subset</a> of the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of <i>n</i>".<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> There is a larger class of number-theoretic functions that do not fit this definition, for example, the <a href="/wiki/Prime-counting_function" title="Prime-counting function">prime-counting functions</a>. This article provides links to functions of both classes. </p><p>An example of an arithmetic function is the <a href="/wiki/Divisor_function" title="Divisor function">divisor function</a> whose value at a positive integer <i>n</i> is equal to the number of divisors of <i>n</i>. </p><p>Arithmetic functions are often extremely irregular (see <a href="#First_100_values_of_some_arithmetic_functions">table</a>), but some of them have series expansions in terms of <a href="/wiki/Ramanujan%27s_sum" title="Ramanujan's sum">Ramanujan's sum</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Multiplicative_and_additive_functions">Multiplicative and additive functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=1" title="Edit section: Multiplicative and additive functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An arithmetic function <i>a</i> is </p> <ul><li><b><a href="/wiki/Completely_additive_function" class="mw-redirect" title="Completely additive function">completely additive</a></b> if <i>a</i>(<i>mn</i>) = <i>a</i>(<i>m</i>) + <i>a</i>(<i>n</i>) for all natural numbers <i>m</i> and <i>n</i>;</li> <li><b><a href="/wiki/Completely_multiplicative_function" title="Completely multiplicative function">completely multiplicative</a></b> if <i>a</i>(<i>mn</i>) = <i>a</i>(<i>m</i>)<i>a</i>(<i>n</i>) for all natural numbers <i>m</i> and <i>n</i>;</li></ul> <p>Two whole numbers <i>m</i> and <i>n</i> are called <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> if their <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> is 1, that is, if there is no <a href="/wiki/Prime_number" title="Prime number">prime number</a> that divides both of them. </p><p>Then an arithmetic function <i>a</i> is </p> <ul><li><b><a href="/wiki/Additive_function" title="Additive function">additive</a></b> if <i>a</i>(<i>mn</i>) = <i>a</i>(<i>m</i>) + <i>a</i>(<i>n</i>) for all coprime natural numbers <i>m</i> and <i>n</i>;</li> <li><b><a href="/wiki/Multiplicative_function" title="Multiplicative function">multiplicative</a></b> if <i>a</i>(<i>mn</i>) = <i>a</i>(<i>m</i>)<i>a</i>(<i>n</i>) for all coprime natural numbers <i>m</i> and <i>n</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=2" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In this article, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{p}f(p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{p}f(p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bd85a3dda31b6726b12a87db448377418a61b69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.157ex; height:3.176ex;" alt="{\textstyle \sum _{p}f(p)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \prod _{p}f(p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \prod _{p}f(p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d3b6e7fad9ebeb1d75a1b70ea85d7a1fbfa93e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.897ex; height:3.176ex;" alt="{\textstyle \prod _{p}f(p)}"></span> mean that the sum or product is over all <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{p}f(p)=f(2)+f(3)+f(5)+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{p}f(p)=f(2)+f(3)+f(5)+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d411139a7e06f7202ce09968633cedee48b9eb6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.093ex; height:5.676ex;" alt="{\displaystyle \sum _{p}f(p)=f(2)+f(3)+f(5)+\cdots }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{p}f(p)=f(2)f(3)f(5)\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{p}f(p)=f(2)f(3)f(5)\cdots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65c81bb4afacb83810f5c7c2ca1eab944c21ccfd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.608ex; height:5.676ex;" alt="{\displaystyle \prod _{p}f(p)=f(2)f(3)f(5)\cdots .}"></span> Similarly, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{p^{k}}f(p^{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{p^{k}}f(p^{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b20f60948d76e70734ed678e94e260efccd5a82e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.106ex; height:3.509ex;" alt="{\textstyle \sum _{p^{k}}f(p^{k})}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \prod _{p^{k}}f(p^{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \prod _{p^{k}}f(p^{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c549506b3394feca47c5dc8156d7e28a08d275" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.846ex; height:3.509ex;" alt="{\textstyle \prod _{p^{k}}f(p^{k})}"></span> mean that the sum or product is over all <a href="/wiki/Prime_power" title="Prime power">prime powers</a> with strictly positive exponent (so <span class="texhtml"><i>k</i> = 0</span> is not included): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{p^{k}}f(p^{k})=\sum _{p}\sum _{k>0}f(p^{k})=f(2)+f(3)+f(4)+f(5)+f(7)+f(8)+f(9)+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </munder> <mi>f</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> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>></mo> <mn>0</mn> </mrow> </munder> <mi>f</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> <mi>f</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>9</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{p^{k}}f(p^{k})=\sum _{p}\sum _{k>0}f(p^{k})=f(2)+f(3)+f(4)+f(5)+f(7)+f(8)+f(9)+\cdots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b82fc04f836894daa1405721b69e7bb955d46556" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:81.508ex; height:6.343ex;" alt="{\displaystyle \sum _{p^{k}}f(p^{k})=\sum _{p}\sum _{k>0}f(p^{k})=f(2)+f(3)+f(4)+f(5)+f(7)+f(8)+f(9)+\cdots .}"></span> </p><p>The notations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{d\mid n}f(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> <mi>f</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{d\mid n}f(d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d887b2813d557e0ece1533421117a5e29358569" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.68ex; height:3.343ex;" alt="{\textstyle \sum _{d\mid n}f(d)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \prod _{d\mid n}f(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> <mi>f</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \prod _{d\mid n}f(d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c0f6726c50025ee307be2442cac214dafd6e208" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.42ex; height:3.343ex;" alt="{\textstyle \prod _{d\mid n}f(d)}"></span> mean that the sum or product is over all positive divisors of <i>n</i>, including 1 and <i>n</i>. For example, if <span class="texhtml"><i>n</i> = 12</span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{d\mid 12}f(d)=f(1)f(2)f(3)f(4)f(6)f(12).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mn>12</mn> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>12</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{d\mid 12}f(d)=f(1)f(2)f(3)f(4)f(6)f(12).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd786ac947fc45b917c504001a5a6cd890c1551b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:38.07ex; height:6.009ex;" alt="{\displaystyle \prod _{d\mid 12}f(d)=f(1)f(2)f(3)f(4)f(6)f(12).}"></span> </p><p>The notations can be combined: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{p\mid n}f(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> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{p\mid n}f(p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f841f75c84dcffa94290defc52a8e8aec89bde9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.601ex; height:3.343ex;" alt="{\textstyle \sum _{p\mid n}f(p)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \prod _{p\mid n}f(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> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \prod _{p\mid n}f(p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bda77cc1126e6e1fdccda73dc229f04566319b06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.341ex; height:3.343ex;" alt="{\textstyle \prod _{p\mid n}f(p)}"></span> mean that the sum or product is over all prime divisors of <i>n</i>. For example, if <i>n</i> = 18, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{p\mid 18}f(p)=f(2)+f(3),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>∣</mo> <mn>18</mn> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{p\mid 18}f(p)=f(2)+f(3),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c05db4b8d09062c2bf5dca0a1c5dc7eed0e0c0b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:23.086ex; height:6.009ex;" alt="{\displaystyle \sum _{p\mid 18}f(p)=f(2)+f(3),}"></span> and similarly <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{p^{k}\mid n}f(p^{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{p^{k}\mid n}f(p^{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/589ed8a50da8252e7632397c470aaf3c9a820eba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.549ex; height:3.509ex;" alt="{\textstyle \sum _{p^{k}\mid n}f(p^{k})}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \prod _{p^{k}\mid n}f(p^{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \prod _{p^{k}\mid n}f(p^{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb3cb47b03c4f3699dc9cc554565748eb55680b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.289ex; height:3.509ex;" alt="{\textstyle \prod _{p^{k}\mid n}f(p^{k})}"></span> mean that the sum or product is over all prime powers dividing <i>n</i>. For example, if <i>n</i> = 24, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{p^{k}\mid 24}f(p^{k})=f(2)f(3)f(4)f(8).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>∣</mo> <mn>24</mn> </mrow> </munder> <mi>f</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> <mi>f</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{p^{k}\mid 24}f(p^{k})=f(2)f(3)f(4)f(8).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9956fc69cdacc721da2961e3ebc1afe9c4bd0932" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; margin-left: -0.063ex; width:30.331ex; height:6.343ex;" alt="{\displaystyle \prod _{p^{k}\mid 24}f(p^{k})=f(2)f(3)f(4)f(8).}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Ω(n),_ω(n),_νp(n)_–_prime_power_decomposition"><span id=".CE.A9.28n.29.2C_.CF.89.28n.29.2C_.CE.BDp.28n.29_.E2.80.93_prime_power_decomposition"></span>Ω(<i>n</i>), <i>ω</i>(<i>n</i>), <i>ν</i><sub><i>p</i></sub>(<i>n</i>) – prime power decomposition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=3" title="Edit section: Ω(n), ω(n), νp(n) – prime power decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a> states that any positive integer <i>n</i> can be represented uniquely as a product of powers of primes: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=p_{1}^{a_{1}}\cdots p_{k}^{a_{k}}}"> <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>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=p_{1}^{a_{1}}\cdots p_{k}^{a_{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3a0b9fa24964d5354f56263c05bd4b3056ac9d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.224ex; height:3.176ex;" alt="{\displaystyle n=p_{1}^{a_{1}}\cdots p_{k}^{a_{k}}}"></span> where <i>p</i><sub>1</sub> < <i>p</i><sub>2</sub> < ... < <i>p</i><sub><i>k</i></sub> are primes and the <i>a<sub>j</sub></i> are positive integers. (1 is given by the empty product.) </p><p>It is often convenient to write this as an infinite product over all the primes, where all but a finite number have a zero exponent. Define the <a href="/wiki/P-adic_valuation" title="P-adic valuation"><i>p</i>-adic valuation</a> <b>ν<sub><i>p</i></sub>(<i>n</i>)</b> to be the exponent of the highest power of the prime <i>p</i> that divides <i>n</i>. That is, if <i>p</i> is one of the <i>p</i><sub><i>i</i></sub> then <i>ν</i><sub><i>p</i></sub>(<i>n</i>) = <i>a</i><sub><i>i</i></sub>, otherwise it is zero. Then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=\prod _{p}p^{\nu _{p}(n)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munder> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=\prod _{p}p^{\nu _{p}(n)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77d7e50082c0f8b02716c89b1c73e0108d12835f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.811ex; height:5.676ex;" alt="{\displaystyle n=\prod _{p}p^{\nu _{p}(n)}.}"></span> </p><p>In terms of the above the <a href="/wiki/Prime_omega_function" title="Prime omega function">prime omega functions</a> ω and Ω are defined by </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.5em;"><i>ω</i>(<i>n</i>) = <i>k</i>,</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"><div class="block-indent" style="padding-left: 1.5em;">Ω(<i>n</i>) = <i>a</i><sub>1</sub> + <i>a</i><sub>2</sub> + ... + <i>a</i><sub><i>k</i></sub>.</div> <p>To avoid repetition, whenever possible formulas for the functions listed in this article are given in terms of <i>n</i> and the corresponding <i>p</i><sub><i>i</i></sub>, <i>a</i><sub><i>i</i></sub>, ω, and Ω. </p> <div class="mw-heading mw-heading2"><h2 id="Multiplicative_functions">Multiplicative functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=4" title="Edit section: Multiplicative functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="σk(n),_τ(n),_d(n)_–_divisor_sums"><span id=".CF.83k.28n.29.2C_.CF.84.28n.29.2C_d.28n.29_.E2.80.93_divisor_sums"></span>σ<sub><i>k</i></sub>(<i>n</i>), τ(<i>n</i>), <i>d</i>(<i>n</i>) – divisor sums</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=5" title="Edit section: σk(n), τ(n), d(n) – divisor sums"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Divisor_function" title="Divisor function">σ<sub><i>k</i></sub>(<i>n</i>)</a></b> is the sum of the <i>k</i>th powers of the positive divisors of <i>n</i>, including 1 and <i>n</i>, where <i>k</i> is a complex number. </p><p><b>σ<sub>1</sub>(<i>n</i>)</b>, the sum of the (positive) divisors of <i>n</i>, is usually denoted by <b>σ(<i>n</i>)</b>. </p><p>Since a positive number to the zero power is one, <b>σ<sub>0</sub>(<i>n</i>)</b> is therefore the number of (positive) divisors of <i>n</i>; it is usually denoted by <b><i>d</i>(<i>n</i>)</b> or <b>τ(<i>n</i>)</b> (for the German <i>Teiler</i> = divisors). </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{k}(n)=\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{(a_{i}+1)k}-1}{p_{i}^{k}-1}}=\prod _{i=1}^{\omega (n)}\left(1+p_{i}^{k}+p_{i}^{2k}+\cdots +p_{i}^{a_{i}k}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>k</mi> </mrow> </msubsup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>k</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{k}(n)=\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{(a_{i}+1)k}-1}{p_{i}^{k}-1}}=\prod _{i=1}^{\omega (n)}\left(1+p_{i}^{k}+p_{i}^{2k}+\cdots +p_{i}^{a_{i}k}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9739c42674f2ed5dd07f0ea9260798b81c1c95b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:59.286ex; height:7.676ex;" alt="{\displaystyle \sigma _{k}(n)=\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{(a_{i}+1)k}-1}{p_{i}^{k}-1}}=\prod _{i=1}^{\omega (n)}\left(1+p_{i}^{k}+p_{i}^{2k}+\cdots +p_{i}^{a_{i}k}\right).}"></span> </p><p>Setting <i>k</i> = 0 in the second product gives <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau (n)=d(n)=(1+a_{1})(1+a_{2})\cdots (1+a_{\omega (n)}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau (n)=d(n)=(1+a_{1})(1+a_{2})\cdots (1+a_{\omega (n)}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3158d8a489a7c7c09233e96a282a20919f80f3e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:45.922ex; height:3.176ex;" alt="{\displaystyle \tau (n)=d(n)=(1+a_{1})(1+a_{2})\cdots (1+a_{\omega (n)}).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="φ(n)_–_Euler_totient_function"><span id=".CF.86.28n.29_.E2.80.93_Euler_totient_function"></span>φ(<i>n</i>) – Euler totient function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=6" title="Edit section: φ(n) – Euler totient function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Euler_totient_function" class="mw-redirect" title="Euler totient function">φ(<i>n</i>)</a></b>, the Euler totient function, is the number of positive integers not greater than <i>n</i> that are coprime to <i>n</i>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)=n\prod _{p\mid n}\left(1-{\frac {1}{p}}\right)=n\left({\frac {p_{1}-1}{p_{1}}}\right)\left({\frac {p_{2}-1}{p_{2}}}\right)\cdots \left({\frac {p_{\omega (n)}-1}{p_{\omega (n)}}}\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> <mi>n</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mfrac> </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)=n\left({\frac {p_{1}-1}{p_{1}}}\right)\left({\frac {p_{2}-1}{p_{2}}}\right)\cdots \left({\frac {p_{\omega (n)}-1}{p_{\omega (n)}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5cba4ec5f47934d146154fd1c21f4be81163e52" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:66.107ex; height:7.343ex;" alt="{\displaystyle \varphi (n)=n\prod _{p\mid n}\left(1-{\frac {1}{p}}\right)=n\left({\frac {p_{1}-1}{p_{1}}}\right)\left({\frac {p_{2}-1}{p_{2}}}\right)\cdots \left({\frac {p_{\omega (n)}-1}{p_{\omega (n)}}}\right).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Jk(n)_–_Jordan_totient_function"><span id="Jk.28n.29_.E2.80.93_Jordan_totient_function"></span>J<sub><i>k</i></sub>(<i>n</i>) – Jordan totient function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=7" title="Edit section: Jk(n) – Jordan totient function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Jordan_totient_function" class="mw-redirect" title="Jordan totient function">J<sub><i>k</i></sub>(<i>n</i>)</a></b>, the Jordan totient function, is the number of <i>k</i>-tuples of positive integers all less than or equal to <i>n</i> that form a coprime (<i>k</i> + 1)-tuple together with <i>n</i>. It is a generalization of Euler's totient, <span class="texhtml">φ(<i>n</i>) = J<sub>1</sub>(<i>n</i>)</span>. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{k}(n)=n^{k}\prod _{p\mid n}\left(1-{\frac {1}{p^{k}}}\right)=n^{k}\left({\frac {p_{1}^{k}-1}{p_{1}^{k}}}\right)\left({\frac {p_{2}^{k}-1}{p_{2}^{k}}}\right)\cdots \left({\frac {p_{\omega (n)}^{k}-1}{p_{\omega (n)}^{k}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <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> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>−</mo> <mn>1</mn> </mrow> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>−</mo> <mn>1</mn> </mrow> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>−</mo> <mn>1</mn> </mrow> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{k}(n)=n^{k}\prod _{p\mid n}\left(1-{\frac {1}{p^{k}}}\right)=n^{k}\left({\frac {p_{1}^{k}-1}{p_{1}^{k}}}\right)\left({\frac {p_{2}^{k}-1}{p_{2}^{k}}}\right)\cdots \left({\frac {p_{\omega (n)}^{k}-1}{p_{\omega (n)}^{k}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f07f2eb5c1448acb7f76cd63914b6de2ba345ffd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:71.467ex; height:8.176ex;" alt="{\displaystyle J_{k}(n)=n^{k}\prod _{p\mid n}\left(1-{\frac {1}{p^{k}}}\right)=n^{k}\left({\frac {p_{1}^{k}-1}{p_{1}^{k}}}\right)\left({\frac {p_{2}^{k}-1}{p_{2}^{k}}}\right)\cdots \left({\frac {p_{\omega (n)}^{k}-1}{p_{\omega (n)}^{k}}}\right).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="μ(n)_–_Möbius_function"><span id=".CE.BC.28n.29_.E2.80.93_M.C3.B6bius_function"></span>μ(<i>n</i>) – Möbius function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=8" title="Edit section: μ(n) – Möbius function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/M%C3%B6bius_function" title="Möbius function">μ(<i>n</i>)</a></b>, the Möbius function, is important because of the <a href="/wiki/M%C3%B6bius_inversion" class="mw-redirect" title="Möbius inversion">Möbius inversion</a> formula. See <a href="#Dirichlet_convolution">Dirichlet convolution</a>, below. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (n)={\begin{cases}(-1)^{\omega (n)}=(-1)^{\Omega (n)}&{\text{if }}\;\omega (n)=\Omega (n)\\0&{\text{if }}\;\omega (n)\neq \Omega (n).\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>n</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> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mspace width="thickmathspace" /> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mspace width="thickmathspace" /> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (n)={\begin{cases}(-1)^{\omega (n)}=(-1)^{\Omega (n)}&{\text{if }}\;\omega (n)=\Omega (n)\\0&{\text{if }}\;\omega (n)\neq \Omega (n).\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08df85401671edb4bbd6982cb67974f30eb235d6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.246ex; height:6.176ex;" alt="{\displaystyle \mu (n)={\begin{cases}(-1)^{\omega (n)}=(-1)^{\Omega (n)}&{\text{if }}\;\omega (n)=\Omega (n)\\0&{\text{if }}\;\omega (n)\neq \Omega (n).\end{cases}}}"></span> </p><p>This implies that μ(1) = 1. (Because Ω(1) = ω(1) = 0.) </p> <div class="mw-heading mw-heading3"><h3 id="τ(n)_–_Ramanujan_tau_function"><span id=".CF.84.28n.29_.E2.80.93_Ramanujan_tau_function"></span>τ(<i>n</i>) – Ramanujan tau function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=9" title="Edit section: τ(n) – Ramanujan tau function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Ramanujan_tau_function" title="Ramanujan tau function">τ(<i>n</i>)</a></b>, the Ramanujan tau function, is defined by its <a href="/wiki/Generating_function" title="Generating function">generating function</a> identity: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n\geq 1}\tau (n)q^{n}=q\prod _{n\geq 1}(1-q^{n})^{24}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </munder> <mi>τ</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> <mo>=</mo> <mi>q</mi> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n\geq 1}\tau (n)q^{n}=q\prod _{n\geq 1}(1-q^{n})^{24}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a6a7ec766c05e9ad9163dd9edcde8e19945d92d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.721ex; height:5.676ex;" alt="{\displaystyle \sum _{n\geq 1}\tau (n)q^{n}=q\prod _{n\geq 1}(1-q^{n})^{24}.}"></span> </p><p>Although it is hard to say exactly what "arithmetical property of <i>n</i>" it "expresses",<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> (<i>τ</i>(<i>n</i>) is (2π)<sup>−12</sup> times the <i>n</i>th Fourier coefficient in the <a href="/wiki/Q-expansion" class="mw-redirect" title="Q-expansion">q-expansion</a> of the <a href="/wiki/Modular_discriminant#Modular_discriminant" class="mw-redirect" title="Modular discriminant">modular discriminant</a> function)<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> it is included among the arithmetical functions because it is multiplicative and it occurs in identities involving certain σ<sub><i>k</i></sub>(<i>n</i>) and <i>r</i><sub><i>k</i></sub>(<i>n</i>) functions (because these are also coefficients in the expansion of <a href="/wiki/Modular_form" title="Modular form">modular forms</a>). </p> <div class="mw-heading mw-heading3"><h3 id="cq(n)_–_Ramanujan's_sum"><span id="cq.28n.29_.E2.80.93_Ramanujan.27s_sum"></span><i>c</i><sub><i>q</i></sub>(<i>n</i>) – Ramanujan's sum</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=10" title="Edit section: cq(n) – Ramanujan's sum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Ramanujan%27s_sum" title="Ramanujan's sum"><i>c</i><sub><i>q</i></sub>(<i>n</i>)</a></b>, Ramanujan's sum, is the sum of the <i>n</i>th powers of the primitive <i>q</i>th <a href="/wiki/Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{q}(n)=\sum _{\stackrel {1\leq a\leq q}{\gcd(a,q)=1}}e^{2\pi i{\tfrac {a}{q}}n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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>a</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>a</mi> <mo>≤</mo> <mi>q</mi> </mrow> </mover> </mrow> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>q</mi> </mfrac> </mstyle> </mrow> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{q}(n)=\sum _{\stackrel {1\leq a\leq q}{\gcd(a,q)=1}}e^{2\pi i{\tfrac {a}{q}}n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eed373ad44220653c398a57da03ef454b32b689" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:23.6ex; height:8.509ex;" alt="{\displaystyle c_{q}(n)=\sum _{\stackrel {1\leq a\leq q}{\gcd(a,q)=1}}e^{2\pi i{\tfrac {a}{q}}n}.}"></span> </p><p>Even though it is defined as a sum of complex numbers (irrational for most values of <i>q</i>), it is an integer. For a fixed value of <i>n</i> it is multiplicative in <i>q</i>: </p> <dl><dd><b>If <i>q</i> and <i>r</i> are coprime</b>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{q}(n)c_{r}(n)=c_{qr}(n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{q}(n)c_{r}(n)=c_{qr}(n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/595a92b25f2f0d803a228b73cd76908e610d6616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.07ex; height:3.009ex;" alt="{\displaystyle c_{q}(n)c_{r}(n)=c_{qr}(n).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="ψ(n)_-_Dedekind_psi_function"><span id=".CF.88.28n.29_-_Dedekind_psi_function"></span><i>ψ</i>(<i>n</i>) - Dedekind psi function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=11" title="Edit section: ψ(n) - Dedekind psi function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Dedekind_psi_function" title="Dedekind psi function">Dedekind psi function</a>, used in the theory of <a href="/wiki/Modular_function" class="mw-redirect" title="Modular function">modular functions</a>, is defined by the formula <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (n)=n\prod _{p|n}\left(1+{\frac {1}{p}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <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 \psi (n)=n\prod _{p|n}\left(1+{\frac {1}{p}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39a3f18eff74513154cd10ecb3d0341bd10b9828" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:23.418ex; height:7.176ex;" alt="{\displaystyle \psi (n)=n\prod _{p|n}\left(1+{\frac {1}{p}}\right).}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Completely_multiplicative_functions">Completely multiplicative functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=12" title="Edit section: Completely multiplicative functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="λ(n)_–_Liouville_function"><span id=".CE.BB.28n.29_.E2.80.93_Liouville_function"></span>λ(<i>n</i>) – Liouville function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=13" title="Edit section: λ(n) – Liouville function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Liouville_function" title="Liouville function"><i>λ</i>(<i>n</i>)</a></b>, the Liouville function, is defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (n)=(-1)^{\Omega (n)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda (n)=(-1)^{\Omega (n)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3314fa23bf14b49769388fb53a406110d653db4f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.769ex; height:3.343ex;" alt="{\displaystyle \lambda (n)=(-1)^{\Omega (n)}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="χ(n)_–_characters"><span id=".CF.87.28n.29_.E2.80.93_characters"></span><i>χ</i>(<i>n</i>) – characters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=14" title="Edit section: χ(n) – characters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All <b><a href="/wiki/Dirichlet_character" title="Dirichlet character">Dirichlet characters</a> <i>χ</i>(<i>n</i>)</b> are completely multiplicative. Two characters have special notations: </p><p>The <b>principal character (mod <i>n</i>)</b> is denoted by <i>χ</i><sub>0</sub>(<i>a</i>) (or <i>χ</i><sub>1</sub>(<i>a</i>)). It is defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{0}(a)={\begin{cases}1&{\text{if }}\gcd(a,n)=1,\\0&{\text{if }}\gcd(a,n)\neq 1.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</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>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>1.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{0}(a)={\begin{cases}1&{\text{if }}\gcd(a,n)=1,\\0&{\text{if }}\gcd(a,n)\neq 1.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae40721f537abbf05c8fe9f11e3ce20a1908cfb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.816ex; height:6.176ex;" alt="{\displaystyle \chi _{0}(a)={\begin{cases}1&{\text{if }}\gcd(a,n)=1,\\0&{\text{if }}\gcd(a,n)\neq 1.\end{cases}}}"></span> </p><p>The <b>quadratic character (mod <i>n</i>)</b> is denoted by the <a href="/wiki/Jacobi_symbol" title="Jacobi symbol">Jacobi symbol</a> for odd <i>n</i> (it is not defined for even <i>n</i>): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {a}{n}}\right)=\left({\frac {a}{p_{1}}}\right)^{a_{1}}\left({\frac {a}{p_{2}}}\right)^{a_{2}}\cdots \left({\frac {a}{p_{\omega (n)}}}\right)^{a_{\omega (n)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>⋯</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {a}{n}}\right)=\left({\frac {a}{p_{1}}}\right)^{a_{1}}\left({\frac {a}{p_{2}}}\right)^{a_{2}}\cdots \left({\frac {a}{p_{\omega (n)}}}\right)^{a_{\omega (n)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e095727b399825cbb14bd3a25c0ef37d7fff337b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.96ex; height:6.343ex;" alt="{\displaystyle \left({\frac {a}{n}}\right)=\left({\frac {a}{p_{1}}}\right)^{a_{1}}\left({\frac {a}{p_{2}}}\right)^{a_{2}}\cdots \left({\frac {a}{p_{\omega (n)}}}\right)^{a_{\omega (n)}}.}"></span> </p><p>In this formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\tfrac {a}{p}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>p</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\tfrac {a}{p}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ebb252decd96b7adf4d3fffdda2c15b718c7420" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.515ex; height:3.343ex;" alt="{\displaystyle ({\tfrac {a}{p}})}"></span> is the <a href="/wiki/Legendre_symbol" title="Legendre symbol">Legendre symbol</a>, defined for all integers <i>a</i> and all odd primes <i>p</i> by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {a}{p}}\right)={\begin{cases}\;\;\,0&{\text{if }}a\equiv 0{\pmod {p}},\\+1&{\text{if }}a\not \equiv 0{\pmod {p}}{\text{ and for some integer }}x,\;a\equiv x^{2}{\pmod {p}}\\-1&{\text{if there is no such }}x.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>a</mi> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>a</mi> <mo>≢</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> and for some integer </mtext> </mrow> <mi>x</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mo>≡</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if there is no such </mtext> </mrow> <mi>x</mi> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {a}{p}}\right)={\begin{cases}\;\;\,0&{\text{if }}a\equiv 0{\pmod {p}},\\+1&{\text{if }}a\not \equiv 0{\pmod {p}}{\text{ and for some integer }}x,\;a\equiv x^{2}{\pmod {p}}\\-1&{\text{if there is no such }}x.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6867a07912d788c45d675284e4be4437ae9d313" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:74.441ex; height:8.843ex;" alt="{\displaystyle \left({\frac {a}{p}}\right)={\begin{cases}\;\;\,0&{\text{if }}a\equiv 0{\pmod {p}},\\+1&{\text{if }}a\not \equiv 0{\pmod {p}}{\text{ and for some integer }}x,\;a\equiv x^{2}{\pmod {p}}\\-1&{\text{if there is no such }}x.\end{cases}}}"></span> </p><p>Following the normal convention for the empty product, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {a}{1}}\right)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mn>1</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {a}{1}}\right)=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab68e93d279e8d82d1153662b5aeb056dbcd50e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.749ex; height:4.843ex;" alt="{\displaystyle \left({\frac {a}{1}}\right)=1.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Additive_functions">Additive functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=15" title="Edit section: Additive functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="ω(n)_–_distinct_prime_divisors"><span id=".CF.89.28n.29_.E2.80.93_distinct_prime_divisors"></span><i>ω</i>(<i>n</i>) – distinct prime divisors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=16" title="Edit section: ω(n) – distinct prime divisors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>ω(<i>n</i>)</b>, defined above as the number of distinct primes dividing <i>n</i>, is additive (see <a href="/wiki/Prime_omega_function" title="Prime omega function">Prime omega function</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Completely_additive_functions">Completely additive functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=17" title="Edit section: Completely additive functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Ω(n)_–_prime_divisors"><span id=".CE.A9.28n.29_.E2.80.93_prime_divisors"></span>Ω(<i>n</i>) – prime divisors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=18" title="Edit section: Ω(n) – prime divisors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">Ω(<i>n</i>)</a></b>, defined above as the number of prime factors of <i>n</i> counted with multiplicities, is completely additive (see <a href="/wiki/Prime_omega_function" title="Prime omega function">Prime omega function</a>). </p> <div class="mw-heading mw-heading3"><h3 id="νp(n)_–_p-adic_valuation_of_an_integer_n"><span id=".CE.BDp.28n.29_.E2.80.93_p-adic_valuation_of_an_integer_n"></span><i>ν</i><sub><i>p</i></sub>(<i>n</i>) – <a href="/wiki/P-adic_valuation" title="P-adic valuation"><i>p</i>-adic valuation</a> of an integer <i>n</i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=19" title="Edit section: νp(n) – p-adic valuation of an integer n"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a fixed prime <i>p</i>, <b><i>ν</i><sub><i>p</i></sub>(<i>n</i>)</b>, defined above as the exponent of the largest power of <i>p</i> dividing <i>n</i>, is completely additive. </p> <div class="mw-heading mw-heading3"><h3 id="Logarithmic_derivative">Logarithmic derivative</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=20" title="Edit section: Logarithmic derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ld} (n)={\frac {D(n)}{n}}=\sum _{\stackrel {p\mid n}{p{\text{ prime}}}}{\frac {v_{p}(n)}{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ld</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> prime</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>∣</mo> <mi>n</mi> </mrow> </mover> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mi>p</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ld} (n)={\frac {D(n)}{n}}=\sum _{\stackrel {p\mid n}{p{\text{ prime}}}}{\frac {v_{p}(n)}{p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9106364adaeeee7c32f691abcdcf476845f0737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:29.441ex; height:9.009ex;" alt="{\displaystyle \operatorname {ld} (n)={\frac {D(n)}{n}}=\sum _{\stackrel {p\mid n}{p{\text{ prime}}}}{\frac {v_{p}(n)}{p}}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c770d316f90c97bc02eb0813fe986be35b37f623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.128ex; height:2.843ex;" alt="{\displaystyle D(n)}"></span> is the arithmetic derivative. </p> <div class="mw-heading mw-heading2"><h2 id="Neither_multiplicative_nor_additive">Neither multiplicative nor additive</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=21" title="Edit section: Neither multiplicative nor additive"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="π(x),_Π(x),_θ(x),_ψ(x)_–_prime-counting_functions"><span id=".CF.80.28x.29.2C_.CE.A0.28x.29.2C_.CE.B8.28x.29.2C_.CF.88.28x.29_.E2.80.93_prime-counting_functions"></span><span class="texhtml mvar" style="font-style:italic;">π</span>(<i>x</i>), Π(<i>x</i>), <i>θ</i>(<i>x</i>), <i>ψ</i>(<i>x</i>) – prime-counting functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=22" title="Edit section: π(x), Π(x), θ(x), ψ(x) – prime-counting functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These important functions (which are not arithmetic functions) are defined for non-negative real arguments, and are used in the various statements and proofs of the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a>. They are summation functions (see the main section just below) of arithmetic functions which are neither multiplicative nor additive. </p><p><span class="texhtml mvar" style="font-style:italic;">π</span>(<i>x</i>), the <a href="/wiki/Prime-counting_function" title="Prime-counting function">prime-counting function</a>, is the number of primes not exceeding <i>x</i>. It is the summation function of the <a href="/wiki/Indicator_function" title="Indicator function">characteristic function</a> of the prime numbers. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (x)=\sum _{p\leq x}1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (x)=\sum _{p\leq x}1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150521416b4ea4c3831f5a737f3ea6315e9b6906" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:12.474ex; height:5.843ex;" alt="{\displaystyle \pi (x)=\sum _{p\leq x}1}"></span> </p><p>A related function counts prime powers with weight 1 for primes, 1/2 for their squares, 1/3 for cubes, etc. It is the summation function of the arithmetic function which takes the value 1/<i>k</i> on integers which are the <i>k</i>-th power of some prime number, and the value 0 on other integers. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi (x)=\sum _{p^{k}\leq x}{\frac {1}{k}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi (x)=\sum _{p^{k}\leq x}{\frac {1}{k}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c395d0c931c9ef52d59d39200fa908a6b68159cb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:14.967ex; height:7.176ex;" alt="{\displaystyle \Pi (x)=\sum _{p^{k}\leq x}{\frac {1}{k}}.}"></span> </p><p><i>ϑ</i>(<i>x</i>) and <i>ψ</i>(<i>x</i>), the <a href="/wiki/Chebyshev_function" title="Chebyshev function">Chebyshev functions</a>, are defined as sums of the natural logarithms of the primes not exceeding <i>x</i>. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta (x)=\sum _{p\leq x}\log p,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϑ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <mi>log</mi> <mo>⁡</mo> <mi>p</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta (x)=\sum _{p\leq x}\log p,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cd0324a266a7d428f678de07cae448f48176f1d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.528ex; height:5.843ex;" alt="{\displaystyle \vartheta (x)=\sum _{p\leq x}\log p,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)=\sum _{p^{k}\leq x}\log p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <mi>log</mi> <mo>⁡</mo> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)=\sum _{p^{k}\leq x}\log p.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab6a98a782ce097018fa23a53ecc48d5224a1c30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:17.218ex; height:6.343ex;" alt="{\displaystyle \psi (x)=\sum _{p^{k}\leq x}\log p.}"></span> </p><p>The second Chebyshev function <i>ψ</i>(<i>x</i>) is the summation function of the von Mangoldt function just below. </p> <div class="mw-heading mw-heading3"><h3 id="Λ(n)_–_von_Mangoldt_function"><span id=".CE.9B.28n.29_.E2.80.93_von_Mangoldt_function"></span>Λ(<i>n</i>) – von Mangoldt function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=23" title="Edit section: Λ(n) – von Mangoldt function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Von_Mangoldt_function" title="Von Mangoldt function">Λ(<i>n</i>)</a></b>, the von Mangoldt function, is 0 unless the argument <i>n</i> is a prime power <span class="texhtml"><i>p</i><sup><i>k</i></sup></span>, in which case it is the natural log of the prime <i>p</i>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (n)={\begin{cases}\log p&{\text{if }}n=2,3,4,5,7,8,9,11,13,16,\ldots =p^{k}{\text{ is a prime power}}\\0&{\text{if }}n=1,6,10,12,14,15,18,20,21,\dots \;\;\;\;{\text{ is not a prime power}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>n</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> <mi>log</mi> <mo>⁡</mo> <mi>p</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>9</mn> <mo>,</mo> <mn>11</mn> <mo>,</mo> <mn>13</mn> <mo>,</mo> <mn>16</mn> <mo>,</mo> <mo>…</mo> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> is a prime power</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>10</mn> <mo>,</mo> <mn>12</mn> <mo>,</mo> <mn>14</mn> <mo>,</mo> <mn>15</mn> <mo>,</mo> <mn>18</mn> <mo>,</mo> <mn>20</mn> <mo>,</mo> <mn>21</mn> <mo>,</mo> <mo>…</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> is not a prime power</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda (n)={\begin{cases}\log p&{\text{if }}n=2,3,4,5,7,8,9,11,13,16,\ldots =p^{k}{\text{ is a prime power}}\\0&{\text{if }}n=1,6,10,12,14,15,18,20,21,\dots \;\;\;\;{\text{ is not a prime power}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f46eaff23407e3564fc538652fff31b915043917" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:78.821ex; height:6.176ex;" alt="{\displaystyle \Lambda (n)={\begin{cases}\log p&{\text{if }}n=2,3,4,5,7,8,9,11,13,16,\ldots =p^{k}{\text{ is a prime power}}\\0&{\text{if }}n=1,6,10,12,14,15,18,20,21,\dots \;\;\;\;{\text{ is not a prime power}}.\end{cases}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="p(n)_–_partition_function"><span id="p.28n.29_.E2.80.93_partition_function"></span><i>p</i>(<i>n</i>) – partition function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=24" title="Edit section: p(n) – partition function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Partition_function_(number_theory)" title="Partition function (number theory)"><i>p</i>(<i>n</i>)</a></b>, the partition function, is the number of ways of representing <i>n</i> as a sum of positive integers, where two representations with the same summands in a different order are not counted as being different: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(n)=\left|\left\{(a_{1},a_{2},\dots a_{k}):0<a_{1}\leq a_{2}\leq \cdots \leq a_{k}\;\land \;n=a_{1}+a_{2}+\cdots +a_{k}\right\}\right|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:</mo> <mn>0</mn> <mo><</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mo>∧</mo> <mspace width="thickmathspace" /> <mi>n</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> <mo>|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(n)=\left|\left\{(a_{1},a_{2},\dots a_{k}):0<a_{1}\leq a_{2}\leq \cdots \leq a_{k}\;\land \;n=a_{1}+a_{2}+\cdots +a_{k}\right\}\right|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f0ba6a694a172227aa18facb1235cf337148bf8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:77.689ex; height:2.843ex;" alt="{\displaystyle p(n)=\left|\left\{(a_{1},a_{2},\dots a_{k}):0<a_{1}\leq a_{2}\leq \cdots \leq a_{k}\;\land \;n=a_{1}+a_{2}+\cdots +a_{k}\right\}\right|.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="λ(n)_–_Carmichael_function"><span id=".CE.BB.28n.29_.E2.80.93_Carmichael_function"></span>λ(<i>n</i>) – Carmichael function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=25" title="Edit section: λ(n) – Carmichael function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Carmichael_function" title="Carmichael function"><i>λ</i>(<i>n</i>)</a></b>, the Carmichael function, is the smallest positive number such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{\lambda (n)}\equiv 1{\pmod {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\lambda (n)}\equiv 1{\pmod {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e475506bcb72caae57599e0ccafa7efe023be9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.026ex; height:3.343ex;" alt="{\displaystyle a^{\lambda (n)}\equiv 1{\pmod {n}}}"></span>   for all <i>a</i> coprime to <i>n</i>. Equivalently, it is the <a href="/wiki/Least_common_multiple" title="Least common multiple">least common multiple</a> of the orders of the elements of the <a href="/wiki/Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">multiplicative group of integers modulo <i>n</i></a>. </p><p>For powers of odd primes and for 2 and 4, <i>λ</i>(<i>n</i>) is equal to the Euler totient function of <i>n</i>; for powers of 2 greater than 4 it is equal to one half of the Euler totient function of <i>n</i>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (n)={\begin{cases}\;\;\phi (n)&{\text{if }}n=2,3,4,5,7,9,11,13,17,19,23,25,27,\dots \\{\tfrac {1}{2}}\phi (n)&{\text{if }}n=8,16,32,64,\dots \end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>n</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> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>9</mn> <mo>,</mo> <mn>11</mn> <mo>,</mo> <mn>13</mn> <mo>,</mo> <mn>17</mn> <mo>,</mo> <mn>19</mn> <mo>,</mo> <mn>23</mn> <mo>,</mo> <mn>25</mn> <mo>,</mo> <mn>27</mn> <mo>,</mo> <mo>…</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>=</mo> <mn>8</mn> <mo>,</mo> <mn>16</mn> <mo>,</mo> <mn>32</mn> <mo>,</mo> <mn>64</mn> <mo>,</mo> <mo>…</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda (n)={\begin{cases}\;\;\phi (n)&{\text{if }}n=2,3,4,5,7,9,11,13,17,19,23,25,27,\dots \\{\tfrac {1}{2}}\phi (n)&{\text{if }}n=8,16,32,64,\dots \end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbcd2fb2391afb3fff6a8425487e185da258a6b2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:64.569ex; height:6.509ex;" alt="{\displaystyle \lambda (n)={\begin{cases}\;\;\phi (n)&{\text{if }}n=2,3,4,5,7,9,11,13,17,19,23,25,27,\dots \\{\tfrac {1}{2}}\phi (n)&{\text{if }}n=8,16,32,64,\dots \end{cases}}}"></span> and for general <i>n</i> it is the least common multiple of λ of each of the prime power factors of <i>n</i>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (p_{1}^{a_{1}}p_{2}^{a_{2}}\dots p_{\omega (n)}^{a_{\omega (n)}})=\operatorname {lcm} [\lambda (p_{1}^{a_{1}}),\;\lambda (p_{2}^{a_{2}}),\dots ,\lambda (p_{\omega (n)}^{a_{\omega (n)}})].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</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>a</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>a</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>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>lcm</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>λ</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>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mi>λ</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>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda (p_{1}^{a_{1}}p_{2}^{a_{2}}\dots p_{\omega (n)}^{a_{\omega (n)}})=\operatorname {lcm} [\lambda (p_{1}^{a_{1}}),\;\lambda (p_{2}^{a_{2}}),\dots ,\lambda (p_{\omega (n)}^{a_{\omega (n)}})].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a328e8f19a1bc0835ea2e412cb2ce6c4d71d4a51" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:54.288ex; height:4.009ex;" alt="{\displaystyle \lambda (p_{1}^{a_{1}}p_{2}^{a_{2}}\dots p_{\omega (n)}^{a_{\omega (n)}})=\operatorname {lcm} [\lambda (p_{1}^{a_{1}}),\;\lambda (p_{2}^{a_{2}}),\dots ,\lambda (p_{\omega (n)}^{a_{\omega (n)}})].}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="h(n)_–_Class_number"><span id="h.28n.29_.E2.80.93_Class_number"></span><i>h</i>(<i>n</i>) – Class number</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=26" title="Edit section: h(n) – Class number"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Ideal_class_group" title="Ideal class group"><i>h</i>(<i>n</i>)</a></b>, the class number function, is the order of the <a href="/wiki/Ideal_class_group" title="Ideal class group">ideal class group</a> of an algebraic extension of the rationals with <a href="/wiki/Discriminant" title="Discriminant">discriminant</a> <i>n</i>. The notation is ambiguous, as there are in general many extensions with the same discriminant. See <a href="/wiki/Quadratic_field" title="Quadratic field">quadratic field</a> and <a href="/wiki/Cyclotomic_field" title="Cyclotomic field">cyclotomic field</a> for classical examples. </p> <div class="mw-heading mw-heading3"><h3 id="rk(n)_–_Sum_of_k_squares"><span id="rk.28n.29_.E2.80.93_Sum_of_k_squares"></span><i>r</i><sub><i>k</i></sub>(<i>n</i>) – Sum of <i>k</i> squares</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=27" title="Edit section: rk(n) – Sum of k squares"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Sum_of_squares_function" title="Sum of squares function"><i>r</i><sub><i>k</i></sub>(<i>n</i>)</a></b> is the number of ways <i>n</i> can be represented as the sum of <i>k</i> squares, where representations that differ only in the order of the summands or in the signs of the square roots are counted as different. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{k}(n)=\left|\left\{(a_{1},a_{2},\dots ,a_{k}):n=a_{1}^{2}+a_{2}^{2}+\cdots +a_{k}^{2}\right\}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:</mo> <mi>n</mi> <mo>=</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>}</mo> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{k}(n)=\left|\left\{(a_{1},a_{2},\dots ,a_{k}):n=a_{1}^{2}+a_{2}^{2}+\cdots +a_{k}^{2}\right\}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4808578f104c7b142cdfeff83ce5d355e83db9dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:51.913ex; height:3.176ex;" alt="{\displaystyle r_{k}(n)=\left|\left\{(a_{1},a_{2},\dots ,a_{k}):n=a_{1}^{2}+a_{2}^{2}+\cdots +a_{k}^{2}\right\}\right|}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="D(n)_–_Arithmetic_derivative"><span id="D.28n.29_.E2.80.93_Arithmetic_derivative"></span><i>D</i>(<i>n</i>) – Arithmetic derivative</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=28" title="Edit section: D(n) – Arithmetic derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using the <a href="/wiki/Differential_operator#Notations" title="Differential operator">Heaviside notation</a> for the derivative, the <a href="/wiki/Arithmetic_derivative" title="Arithmetic derivative">arithmetic derivative</a> <i>D</i>(<i>n</i>) is a function such that </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(n)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(n)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/533ce40a13f694e6b4a4440e252a6dbf8dd27e6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.389ex; height:2.843ex;" alt="{\displaystyle D(n)=1}"></span> if <i>n</i> prime, and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(mn)=mD(n)+D(m)n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> <mi>D</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(mn)=mD(n)+D(m)n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4693d4e5c85471a85a6f7129fae02722d8f790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.445ex; height:2.843ex;" alt="{\displaystyle D(mn)=mD(n)+D(m)n}"></span> (the <a href="/wiki/Product_rule" title="Product rule">product rule</a>)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Summation_functions">Summation functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=29" title="Edit section: Summation functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given an arithmetic function <i>a</i>(<i>n</i>), its <b>summation function</b> <i>A</i>(<i>x</i>) is defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x):=\sum _{n\leq x}a(n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <mi>a</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(x):=\sum _{n\leq x}a(n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc586a47cfea611d12b37f8be85070b6086ddfa2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.45ex; height:5.676ex;" alt="{\displaystyle A(x):=\sum _{n\leq x}a(n).}"></span> <i>A</i> can be regarded as a function of a real variable. Given a positive integer <i>m</i>, <i>A</i> is constant along <a href="/wiki/Open_interval" class="mw-redirect" title="Open interval">open intervals</a> <i>m</i> < <i>x</i> < <i>m</i> + 1, and has a <a href="/wiki/Classification_of_discontinuities" title="Classification of discontinuities">jump discontinuity</a> at each integer for which <i>a</i>(<i>m</i>) ≠ 0. </p><p>Since such functions are often represented by series and integrals, to achieve pointwise convergence it is usual to define the value at the discontinuities as the average of the values to the left and right: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{0}(m):={\frac {1}{2}}\left(\sum _{n<m}a(n)+\sum _{n\leq m}a(n)\right)=A(m)-{\frac {1}{2}}a(m).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo><</mo> <mi>m</mi> </mrow> </munder> <mi>a</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≤</mo> <mi>m</mi> </mrow> </munder> <mi>a</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}(m):={\frac {1}{2}}\left(\sum _{n<m}a(n)+\sum _{n\leq m}a(n)\right)=A(m)-{\frac {1}{2}}a(m).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4e08f71f061117e3e2d28681476c63c82624820" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.613ex; height:7.509ex;" alt="{\displaystyle A_{0}(m):={\frac {1}{2}}\left(\sum _{n<m}a(n)+\sum _{n\leq m}a(n)\right)=A(m)-{\frac {1}{2}}a(m).}"></span> </p><p>Individual values of arithmetic functions may fluctuate wildly – as in most of the above examples. Summation functions "smooth out" these fluctuations. In some cases it may be possible to find <a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">asymptotic behaviour</a> for the summation function for large <i>x</i>. </p><p>A classical example of this phenomenon<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> is given by the <a href="/wiki/Divisor_summatory_function" title="Divisor summatory function">divisor summatory function</a>, the summation function of <i>d</i>(<i>n</i>), the number of divisors of <i>n</i>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \liminf _{n\to \infty }d(n)=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \liminf _{n\to \infty }d(n)=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dff825c017ae5dc5776b6c8139575aa930145933" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.337ex; height:3.843ex;" alt="{\displaystyle \liminf _{n\to \infty }d(n)=2}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \limsup _{n\to \infty }{\frac {\log d(n)\log \log n}{\log n}}=\log 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \limsup _{n\to \infty }{\frac {\log d(n)\log \log n}{\log n}}=\log 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/244055680639c43f27b25a8229a82199a71eaff9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.24ex; height:6.176ex;" alt="{\displaystyle \limsup _{n\to \infty }{\frac {\log d(n)\log \log n}{\log n}}=\log 2}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {d(1)+d(2)+\cdots +d(n)}{\log(1)+\log(2)+\cdots +\log(n)}}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {d(1)+d(2)+\cdots +d(n)}{\log(1)+\log(2)+\cdots +\log(n)}}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b88cb6460f577c119d66247e17ec77fcbaa4377" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.711ex; height:6.509ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {d(1)+d(2)+\cdots +d(n)}{\log(1)+\log(2)+\cdots +\log(n)}}=1.}"></span> </p><p>An <b><a href="/wiki/Average_order_of_an_arithmetic_function" title="Average order of an arithmetic function">average order of an arithmetic function</a></b> is some simpler or better-understood function which has the same summation function asymptotically, and hence takes the same values "on average". We say that <i>g</i> is an <i>average order</i> of <i>f</i> if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n\leq x}f(n)\sim \sum _{n\leq x}g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n\leq x}f(n)\sim \sum _{n\leq x}g(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11f3598e9d3ea8ebff694330ed5abadeae9b7aa5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.385ex; height:5.676ex;" alt="{\displaystyle \sum _{n\leq x}f(n)\sim \sum _{n\leq x}g(n)}"></span> </p><p>as <i>x</i> tends to infinity. The example above shows that <i>d</i>(<i>n</i>) has the average order log(<i>n</i>).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Dirichlet_convolution">Dirichlet convolution</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=30" title="Edit section: Dirichlet convolution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given an arithmetic function <i>a</i>(<i>n</i>), let <i>F</i><sub><i>a</i></sub>(<i>s</i>), for complex <i>s</i>, be the function defined by the corresponding <a href="/wiki/Dirichlet_series" title="Dirichlet series">Dirichlet series</a> (where it <a href="/wiki/Convergent_series" title="Convergent series">converges</a>):<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{a}(s):=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{a}(s):=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5e51ff2b7fc0444d544c15bb85d1e97be7ebdf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.9ex; height:6.843ex;" alt="{\displaystyle F_{a}(s):=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}}.}"></span> <i>F</i><sub><i>a</i></sub>(<i>s</i>) is called a <a href="/wiki/Generating_function" title="Generating function">generating function</a> of <i>a</i>(<i>n</i>). The simplest such series, corresponding to the constant function <i>a</i>(<i>n</i>) = 1 for all <i>n</i>, is <i>ζ</i>(<i>s</i>) the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>. </p><p>The generating function of the Möbius function is the inverse of the zeta function: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)\,\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}=1,\;\;\Re s>1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi mathvariant="normal">ℜ</mi> <mi>s</mi> <mo>></mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)\,\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}=1,\;\;\Re s>1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e2f4e828c1cf4592e2935347affa9170bc708bf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.461ex; height:6.843ex;" alt="{\displaystyle \zeta (s)\,\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}=1,\;\;\Re s>1.}"></span> </p><p>Consider two arithmetic functions <i>a</i> and <i>b</i> and their respective generating functions <i>F</i><sub><i>a</i></sub>(<i>s</i>) and <i>F</i><sub><i>b</i></sub>(<i>s</i>). The product <i>F</i><sub><i>a</i></sub>(<i>s</i>)<i>F</i><sub><i>b</i></sub>(<i>s</i>) can be computed as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{a}(s)F_{b}(s)=\left(\sum _{m=1}^{\infty }{\frac {a(m)}{m^{s}}}\right)\left(\sum _{n=1}^{\infty }{\frac {b(n)}{n^{s}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</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>a</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</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> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{a}(s)F_{b}(s)=\left(\sum _{m=1}^{\infty }{\frac {a(m)}{m^{s}}}\right)\left(\sum _{n=1}^{\infty }{\frac {b(n)}{n^{s}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccaca41200e7117014b5b51265d5aae0c4b1b98a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.336ex; height:7.509ex;" alt="{\displaystyle F_{a}(s)F_{b}(s)=\left(\sum _{m=1}^{\infty }{\frac {a(m)}{m^{s}}}\right)\left(\sum _{n=1}^{\infty }{\frac {b(n)}{n^{s}}}\right).}"></span> </p><p>It is a straightforward exercise to show that if <i>c</i>(<i>n</i>) is defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(n):=\sum _{ij=n}a(i)b(j)=\sum _{i\mid n}a(i)b\left({\frac {n}{i}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mo>=</mo> <mi>n</mi> </mrow> </munder> <mi>a</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>a</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mi>b</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>i</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(n):=\sum _{ij=n}a(i)b(j)=\sum _{i\mid n}a(i)b\left({\frac {n}{i}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c14e9c4d74e4f27e8ce840a52ab215b605dee1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:37.566ex; height:6.509ex;" alt="{\displaystyle c(n):=\sum _{ij=n}a(i)b(j)=\sum _{i\mid n}a(i)b\left({\frac {n}{i}}\right),}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{c}(s)=F_{a}(s)F_{b}(s).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{c}(s)=F_{a}(s)F_{b}(s).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b19c1b074ade6b214187403000ff7b9e8c313df7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.912ex; height:2.843ex;" alt="{\displaystyle F_{c}(s)=F_{a}(s)F_{b}(s).}"></span> </p><p>This function <i>c</i> is called the <a href="/wiki/Dirichlet_convolution" title="Dirichlet convolution">Dirichlet convolution</a> of <i>a</i> and <i>b</i>, and is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a*b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∗</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a*b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4baddb0eb2fc85a456316d026699d38f5166a27c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.422ex; height:2.176ex;" alt="{\displaystyle a*b}"></span>. </p><p>A particularly important case is convolution with the constant function <i>a</i>(<i>n</i>) = 1 for all <i>n</i>, corresponding to multiplying the generating function by the zeta function: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n)=\sum _{d\mid n}f(d).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</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>f</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(n)=\sum _{d\mid n}f(d).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d7784af06911fed2c39cb164cf927586530791" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:16.111ex; height:6.009ex;" alt="{\displaystyle g(n)=\sum _{d\mid n}f(d).}"></span> </p><p>Multiplying by the inverse of the zeta function gives the <a href="/wiki/M%C3%B6bius_inversion" class="mw-redirect" title="Möbius inversion">Möbius inversion</a> formula: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=\sum _{d\mid n}\mu \left({\frac {n}{d}}\right)g(d).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=\sum _{d\mid n}\mu \left({\frac {n}{d}}\right)g(d).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b23bd4a633c52920404cd8bd52ff11a22595a31" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:23.293ex; height:6.509ex;" alt="{\displaystyle f(n)=\sum _{d\mid n}\mu \left({\frac {n}{d}}\right)g(d).}"></span> </p><p>If <i>f</i> is multiplicative, then so is <i>g</i>. If <i>f</i> is completely multiplicative, then <i>g</i> is multiplicative, but may or may not be completely multiplicative. </p> <div class="mw-heading mw-heading2"><h2 id="Relations_among_the_functions">Relations among the functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=31" title="Edit section: Relations among the functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are a great many formulas connecting arithmetical functions with each other and with the functions of analysis, especially powers, roots, and the exponential and log functions. The page <a href="/wiki/Divisor_sum_identities" title="Divisor sum identities">divisor sum identities</a> contains many more generalized and related examples of identities involving arithmetic functions. </p><p>Here are a few examples: </p> <div class="mw-heading mw-heading3"><h3 id="Dirichlet_convolutions">Dirichlet convolutions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=32" title="Edit section: Dirichlet convolutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\delta \mid n}\mu (\delta )=\sum _{\delta \mid n}\lambda \left({\frac {n}{\delta }}\right)|\mu (\delta )|={\begin{cases}1&{\text{if }}n=1\\0&{\text{if }}n\neq 1\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>λ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>δ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≠</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\delta \mid n}\mu (\delta )=\sum _{\delta \mid n}\lambda \left({\frac {n}{\delta }}\right)|\mu (\delta )|={\begin{cases}1&{\text{if }}n=1\\0&{\text{if }}n\neq 1\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92e0d45e029769bb70bf1b00ce7585cf80df68a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:44.204ex; height:7.176ex;" alt="{\displaystyle \sum _{\delta \mid n}\mu (\delta )=\sum _{\delta \mid n}\lambda \left({\frac {n}{\delta }}\right)|\mu (\delta )|={\begin{cases}1&{\text{if }}n=1\\0&{\text{if }}n\neq 1\end{cases}}}"></span>     where <i>λ</i> is the Liouville function.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\delta \mid n}\varphi (\delta )=n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\delta \mid n}\varphi (\delta )=n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4747cb524ca27051585361750fd2f475018ae6e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:13.26ex; height:6.009ex;" alt="{\displaystyle \sum _{\delta \mid n}\varphi (\delta )=n.}"></span>      <sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)\delta =n\sum _{\delta \mid n}{\frac {\mu (\delta )}{\delta }}.}"> <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>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>μ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>δ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <mo>=</mo> <mi>n</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>δ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)\delta =n\sum _{\delta \mid n}{\frac {\mu (\delta )}{\delta }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd7ed8fc4415cfb18c568039a0c31a4817e6123" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:34.161ex; height:7.343ex;" alt="{\displaystyle \varphi (n)=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)\delta =n\sum _{\delta \mid n}{\frac {\mu (\delta )}{\delta }}.}"></span>       Möbius inversion</dd></dl></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{d\mid n}J_{k}(d)=n^{k}.}"> <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> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{d\mid n}J_{k}(d)=n^{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee52b935c557a3a42d067e19663f04956554fcb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.375ex; height:6.009ex;" alt="{\displaystyle \sum _{d\mid n}J_{k}(d)=n^{k}.}"></span>      <sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{k}(n)=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)\delta ^{k}=n^{k}\sum _{\delta \mid n}{\frac {\mu (\delta )}{\delta ^{k}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>μ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>δ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{k}(n)=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)\delta ^{k}=n^{k}\sum _{\delta \mid n}{\frac {\mu (\delta )}{\delta ^{k}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f70c931813907149c78ac25d7ed9346987778c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:37.202ex; height:7.343ex;" alt="{\displaystyle J_{k}(n)=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)\delta ^{k}=n^{k}\sum _{\delta \mid n}{\frac {\mu (\delta )}{\delta ^{k}}}.}"></span>       Möbius inversion</dd></dl></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\delta \mid n}\delta ^{s}J_{r}(\delta )J_{s}\left({\frac {n}{\delta }}\right)=J_{r+s}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>δ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>+</mo> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\delta \mid n}\delta ^{s}J_{r}(\delta )J_{s}\left({\frac {n}{\delta }}\right)=J_{r+s}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb8c2db0c42fa862a0af3d3e38b9ded07d93744d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:29.224ex; height:6.509ex;" alt="{\displaystyle \sum _{\delta \mid n}\delta ^{s}J_{r}(\delta )J_{s}\left({\frac {n}{\delta }}\right)=J_{r+s}(n)}"></span>      <sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\delta \mid n}\varphi (\delta )d\left({\frac {n}{\delta }}\right)=\sigma (n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>δ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\delta \mid n}\varphi (\delta )d\left({\frac {n}{\delta }}\right)=\sigma (n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a525e146f9b24dbc2cb6a06f40b94a2294d0f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:23.008ex; height:6.509ex;" alt="{\displaystyle \sum _{\delta \mid n}\varphi (\delta )d\left({\frac {n}{\delta }}\right)=\sigma (n).}"></span>      <sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\delta \mid n}|\mu (\delta )|=2^{\omega (n)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\delta \mid n}|\mu (\delta )|=2^{\omega (n)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a189c4a8b0dab99c54bc4280c264411d73ffa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:17.723ex; height:6.009ex;" alt="{\displaystyle \sum _{\delta \mid n}|\mu (\delta )|=2^{\omega (n)}.}"></span>      <sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mu (n)|=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)2^{\omega (\delta )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>μ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>δ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mu (n)|=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)2^{\omega (\delta )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ce1639e9cea2089b01979b3a1e6516993bfa06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.007ex; height:6.509ex;" alt="{\displaystyle |\mu (n)|=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)2^{\omega (\delta )}.}"></span>       Möbius inversion</dd></dl></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\delta \mid n}2^{\omega (\delta )}=d(n^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\delta \mid n}2^{\omega (\delta )}=d(n^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0498744ecde08bb3bb17ee1c0ed1c74ab2033033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:17.399ex; height:6.009ex;" alt="{\displaystyle \sum _{\delta \mid n}2^{\omega (\delta )}=d(n^{2}).}"></span>       <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\omega (n)}=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)d(\delta ^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>μ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>δ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\omega (n)}=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)d(\delta ^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03747222207859a9a16cedf7b848d3e6f64cc0f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:24.485ex; height:6.509ex;" alt="{\displaystyle 2^{\omega (n)}=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)d(\delta ^{2}).}"></span>       Möbius inversion</dd></dl></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\delta \mid n}d(\delta ^{2})=d^{2}(n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\delta \mid n}d(\delta ^{2})=d^{2}(n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9778b40f0cb5d0fe110dc243b36c787d9ef08872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:18.097ex; height:6.009ex;" alt="{\displaystyle \sum _{\delta \mid n}d(\delta ^{2})=d^{2}(n).}"></span>       <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(n^{2})=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)d^{2}(\delta ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>μ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>δ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(n^{2})=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)d^{2}(\delta ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34891e7581650786b5f374f5df94cdbd64c6028e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.274ex; height:6.509ex;" alt="{\displaystyle d(n^{2})=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)d^{2}(\delta ).}"></span>       Möbius inversion</dd></dl></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\delta \mid n}d\left({\frac {n}{\delta }}\right)2^{\omega (\delta )}=d^{2}(n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>d</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>δ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\delta \mid n}d\left({\frac {n}{\delta }}\right)2^{\omega (\delta )}=d^{2}(n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30d21683269fc00e2f9a1fbaeddc8056ae1af6b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:24.398ex; height:6.509ex;" alt="{\displaystyle \sum _{\delta \mid n}d\left({\frac {n}{\delta }}\right)2^{\omega (\delta )}=d^{2}(n).}"></span>      </dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\delta \mid n}\lambda (\delta )={\begin{cases}&1{\text{ if }}n{\text{ is a square }}\\&0{\text{ if }}n{\text{ is not square.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>δ</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 /> <mtd> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is a square </mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is not square.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\delta \mid n}\lambda (\delta )={\begin{cases}&1{\text{ if }}n{\text{ is a square }}\\&0{\text{ if }}n{\text{ is not square.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1af9b94306343af862e481d6dfc35d9bfc39ad74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:35.383ex; height:7.176ex;" alt="{\displaystyle \sum _{\delta \mid n}\lambda (\delta )={\begin{cases}&1{\text{ if }}n{\text{ is a square }}\\&0{\text{ if }}n{\text{ is not square.}}\end{cases}}}"></span>     where λ is the <a href="/wiki/Liouville_function" title="Liouville function">Liouville function</a>.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\delta \mid n}\Lambda (\delta )=\log n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\delta \mid n}\Lambda (\delta )=\log n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/012125cf76df22198a69c2bdb04aee25633e797a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:16.712ex; height:6.009ex;" alt="{\displaystyle \sum _{\delta \mid n}\Lambda (\delta )=\log n.}"></span>      <sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (n)=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)\log(\delta ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>μ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>δ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda (n)=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)\log(\delta ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daab588acc3657cd0ea29db2037fa505cdb0844f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.316ex; height:6.509ex;" alt="{\displaystyle \Lambda (n)=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)\log(\delta ).}"></span>       Möbius inversion</dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Sums_of_squares">Sums of squares</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=33" title="Edit section: Sums of squares"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 4,\;\;\;r_{k}(n)>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>4</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 4,\;\;\;r_{k}(n)>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/269ac99fb630e3d8788c4b686e91a191945264fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.691ex; height:2.843ex;" alt="{\displaystyle k\geq 4,\;\;\;r_{k}(n)>0.}"></span>     (<a href="/wiki/Lagrange%27s_four-square_theorem" title="Lagrange's four-square theorem">Lagrange's four-square theorem</a>). </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}(n)=4\sum _{d\mid n}\left({\frac {-4}{d}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>4</mn> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}(n)=4\sum _{d\mid n}\left({\frac {-4}{d}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7a257b4a5ab77994423a51b6c01940a694be7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:21.959ex; height:7.176ex;" alt="{\displaystyle r_{2}(n)=4\sum _{d\mid n}\left({\frac {-4}{d}}\right),}"></span> <sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></dd></dl> <p>where the <a href="/wiki/Kronecker_symbol" title="Kronecker symbol">Kronecker symbol</a> has the values </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {-4}{n}}\right)={\begin{cases}+1&{\text{if }}n\equiv 1{\pmod {4}}\\-1&{\text{if }}n\equiv 3{\pmod {4}}\\\;\;\;0&{\text{if }}n{\text{ is even}}.\\\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>4</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is even</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {-4}{n}}\right)={\begin{cases}+1&{\text{if }}n\equiv 1{\pmod {4}}\\-1&{\text{if }}n\equiv 3{\pmod {4}}\\\;\;\;0&{\text{if }}n{\text{ is even}}.\\\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11735759c3837e661938c47d6f4178d6231ac878" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:35.715ex; height:8.509ex;" alt="{\displaystyle \left({\frac {-4}{n}}\right)={\begin{cases}+1&{\text{if }}n\equiv 1{\pmod {4}}\\-1&{\text{if }}n\equiv 3{\pmod {4}}\\\;\;\;0&{\text{if }}n{\text{ is even}}.\\\end{cases}}}"></span></dd></dl> <p>There is a formula for r<sub>3</sub> in the section on <a href="#Class_number_related">class numbers</a> below. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{4}(n)=8\sum _{\stackrel {d\mid n}{4\,\nmid \,d}}d=8(2+(-1)^{n})\sum _{\stackrel {d\mid n}{2\,\nmid \,d}}d={\begin{cases}8\sigma (n)&{\text{if }}n{\text{ is odd }}\\24\sigma \left({\frac {n}{2^{\nu }}}\right)&{\text{if }}n{\text{ is even }}\end{cases}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>8</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mn>4</mn> <mspace width="thinmathspace" /> <mo>∤</mo> <mspace width="thinmathspace" /> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mi>n</mi> </mrow> </mover> </mrow> </mrow> </munder> <mi>d</mi> <mo>=</mo> <mn>8</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mn>2</mn> <mspace width="thinmathspace" /> <mo>∤</mo> <mspace width="thinmathspace" /> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mi>n</mi> </mrow> </mover> </mrow> </mrow> </munder> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>8</mn> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is odd </mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>24</mn> <mi>σ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is even </mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{4}(n)=8\sum _{\stackrel {d\mid n}{4\,\nmid \,d}}d=8(2+(-1)^{n})\sum _{\stackrel {d\mid n}{2\,\nmid \,d}}d={\begin{cases}8\sigma (n)&{\text{if }}n{\text{ is odd }}\\24\sigma \left({\frac {n}{2^{\nu }}}\right)&{\text{if }}n{\text{ is even }}\end{cases}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5e118ebc2fee36db5bff468d2f26ddba74a1005" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:65.593ex; height:9.676ex;" alt="{\displaystyle r_{4}(n)=8\sum _{\stackrel {d\mid n}{4\,\nmid \,d}}d=8(2+(-1)^{n})\sum _{\stackrel {d\mid n}{2\,\nmid \,d}}d={\begin{cases}8\sigma (n)&{\text{if }}n{\text{ is odd }}\\24\sigma \left({\frac {n}{2^{\nu }}}\right)&{\text{if }}n{\text{ is even }}\end{cases}},}"></span> where <span class="texhtml"><i>ν</i> = <i>ν</i><sub>2</sub>(<i>n</i>)</span>.    <sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{6}(n)=16\sum _{d\mid n}\chi \left({\frac {n}{d}}\right)d^{2}-4\sum _{d\mid n}\chi (d)d^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>16</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>χ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{6}(n)=16\sum _{d\mid n}\chi \left({\frac {n}{d}}\right)d^{2}-4\sum _{d\mid n}\chi (d)d^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67614406f6f04e98b86d5a349e430101a0d7d636" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:39.899ex; height:6.509ex;" alt="{\displaystyle r_{6}(n)=16\sum _{d\mid n}\chi \left({\frac {n}{d}}\right)d^{2}-4\sum _{d\mid n}\chi (d)d^{2},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (n)=\left({\frac {-4}{n}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>4</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi (n)=\left({\frac {-4}{n}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5fd20a9c253ad6e121c59cb1419ed35a591cf0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.019ex; height:6.176ex;" alt="{\displaystyle \chi (n)=\left({\frac {-4}{n}}\right).}"></span><sup id="cite_ref-Hardy_&_Wright,_§_20.13_24-0" class="reference"><a href="#cite_note-Hardy_&_Wright,_§_20.13-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>Define the function <span class="texhtml"><i>σ</i><sub><i>k</i></sub><sup>*</sup>(<i>n</i>)</span> as<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{k}^{*}(n)=(-1)^{n}\sum _{d\mid n}(-1)^{d}d^{k}={\begin{cases}\sum _{d\mid n}d^{k}=\sigma _{k}(n)&{\text{if }}n{\text{ is odd }}\\\sum _{\stackrel {d\mid n}{2\,\mid \,d}}d^{k}-\sum _{\stackrel {d\mid n}{2\,\nmid \,d}}d^{k}&{\text{if }}n{\text{ is even}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is odd </mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mn>2</mn> <mspace width="thinmathspace" /> <mo>∣</mo> <mspace width="thinmathspace" /> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mi>n</mi> </mrow> </mover> </mrow> </mrow> </munder> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mn>2</mn> <mspace width="thinmathspace" /> <mo>∤</mo> <mspace width="thinmathspace" /> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∣</mo> <mi>n</mi> </mrow> </mover> </mrow> </mrow> </munder> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is even</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{k}^{*}(n)=(-1)^{n}\sum _{d\mid n}(-1)^{d}d^{k}={\begin{cases}\sum _{d\mid n}d^{k}=\sigma _{k}(n)&{\text{if }}n{\text{ is odd }}\\\sum _{\stackrel {d\mid n}{2\,\mid \,d}}d^{k}-\sum _{\stackrel {d\mid n}{2\,\nmid \,d}}d^{k}&{\text{if }}n{\text{ is even}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7261e1ab643618d2f3b6d6aec43a41da12ca1c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:65.593ex; height:8.509ex;" alt="{\displaystyle \sigma _{k}^{*}(n)=(-1)^{n}\sum _{d\mid n}(-1)^{d}d^{k}={\begin{cases}\sum _{d\mid n}d^{k}=\sigma _{k}(n)&{\text{if }}n{\text{ is odd }}\\\sum _{\stackrel {d\mid n}{2\,\mid \,d}}d^{k}-\sum _{\stackrel {d\mid n}{2\,\nmid \,d}}d^{k}&{\text{if }}n{\text{ is even}}.\end{cases}}}"></span> </p><p>That is, if <i>n</i> is odd, <span class="texhtml"><i>σ</i><sub><i>k</i></sub><sup>*</sup>(<i>n</i>)</span> is the sum of the <i>k</i>th powers of the divisors of <i>n</i>, that is, <span class="texhtml"><i>σ</i><sub><i>k</i></sub>(<i>n</i>),</span> and if <i>n</i> is even it is the sum of the <i>k</i>th powers of the even divisors of <i>n</i> minus the sum of the <i>k</i>th powers of the odd divisors of <i>n</i>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{8}(n)=16\sigma _{3}^{*}(n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>16</mn> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{8}(n)=16\sigma _{3}^{*}(n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb5a38accb4a48b27b02ad4332b58fea946340ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.966ex; height:3.009ex;" alt="{\displaystyle r_{8}(n)=16\sigma _{3}^{*}(n).}"></span>    <sup id="cite_ref-Hardy_&_Wright,_§_20.13_24-1" class="reference"><a href="#cite_note-Hardy_&_Wright,_§_20.13-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Adopt the convention that Ramanujan's <span class="texhtml"><i>τ</i>(<i>x</i>) = 0</span> if <i>x</i> is <b>not an integer.</b> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{24}(n)={\frac {16}{691}}\sigma _{11}^{*}(n)+{\frac {128}{691}}\left\{(-1)^{n-1}259\tau (n)-512\tau \left({\frac {n}{2}}\right)\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>16</mn> <mn>691</mn> </mfrac> </mrow> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>128</mn> <mn>691</mn> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mn>259</mn> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>512</mn> <mi>τ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{24}(n)={\frac {16}{691}}\sigma _{11}^{*}(n)+{\frac {128}{691}}\left\{(-1)^{n-1}259\tau (n)-512\tau \left({\frac {n}{2}}\right)\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/368f517b29e9a55b6a794e87448326cf681f0be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:59.525ex; height:5.176ex;" alt="{\displaystyle r_{24}(n)={\frac {16}{691}}\sigma _{11}^{*}(n)+{\frac {128}{691}}\left\{(-1)^{n-1}259\tau (n)-512\tau \left({\frac {n}{2}}\right)\right\}}"></span>    <sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Divisor_sum_convolutions">Divisor sum convolutions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=34" title="Edit section: Divisor sum convolutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Here "convolution" does not mean "Dirichlet convolution" but instead refers to the formula for the coefficients of the <a href="/wiki/Power_series#Multiplication_and_division" title="Power series">product of two power series</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\sum _{n=0}^{\infty }a_{n}x^{n}\right)\left(\sum _{n=0}^{\infty }b_{n}x^{n}\right)=\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }a_{i}b_{j}x^{i+j}=\sum _{n=0}^{\infty }\left(\sum _{i=0}^{n}a_{i}b_{n-i}\right)x^{n}=\sum _{n=0}^{\infty }c_{n}x^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\sum _{n=0}^{\infty }a_{n}x^{n}\right)\left(\sum _{n=0}^{\infty }b_{n}x^{n}\right)=\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }a_{i}b_{j}x^{i+j}=\sum _{n=0}^{\infty }\left(\sum _{i=0}^{n}a_{i}b_{n-i}\right)x^{n}=\sum _{n=0}^{\infty }c_{n}x^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c42331b3594b2fc84929158ecb40abc3b01b8d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:79.151ex; height:7.676ex;" alt="{\displaystyle \left(\sum _{n=0}^{\infty }a_{n}x^{n}\right)\left(\sum _{n=0}^{\infty }b_{n}x^{n}\right)=\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }a_{i}b_{j}x^{i+j}=\sum _{n=0}^{\infty }\left(\sum _{i=0}^{n}a_{i}b_{n-i}\right)x^{n}=\sum _{n=0}^{\infty }c_{n}x^{n}.}"></span></dd></dl> <p>The sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}=\sum _{i=0}^{n}a_{i}b_{n-i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}=\sum _{i=0}^{n}a_{i}b_{n-i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4f8c28943bb2755333b7664fec446e8ef9b198" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.157ex; height:6.843ex;" alt="{\displaystyle c_{n}=\sum _{i=0}^{n}a_{i}b_{n-i}}"></span> is called the <a href="/wiki/Convolution" title="Convolution">convolution</a> or the <a href="/wiki/Cauchy_product" title="Cauchy product">Cauchy product</a> of the sequences <i>a</i><sub><i>n</i></sub> and <i>b</i><sub><i>n</i></sub>. <br />These formulas may be proved analytically (see <a href="/wiki/Eisenstein_series" title="Eisenstein series">Eisenstein series</a>) or by elementary methods.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{3}(n)={\frac {1}{5}}\left\{6n\sigma _{1}(n)-\sigma _{1}(n)+12\sum _{0<k<n}\sigma _{1}(k)\sigma _{1}(n-k)\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow> <mn>6</mn> <mi>n</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>12</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{3}(n)={\frac {1}{5}}\left\{6n\sigma _{1}(n)-\sigma _{1}(n)+12\sum _{0<k<n}\sigma _{1}(k)\sigma _{1}(n-k)\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29fc43a8a78eae6518a69965bd29f51bb3f24f8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:58.619ex; height:7.509ex;" alt="{\displaystyle \sigma _{3}(n)={\frac {1}{5}}\left\{6n\sigma _{1}(n)-\sigma _{1}(n)+12\sum _{0<k<n}\sigma _{1}(k)\sigma _{1}(n-k)\right\}.}"></span>    <sup id="cite_ref-Ramanujan,_p._146_29-0" class="reference"><a href="#cite_note-Ramanujan,_p._146-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{5}(n)={\frac {1}{21}}\left\{10(3n-1)\sigma _{3}(n)+\sigma _{1}(n)+240\sum _{0<k<n}\sigma _{1}(k)\sigma _{3}(n-k)\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>21</mn> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow> <mn>10</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>240</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{5}(n)={\frac {1}{21}}\left\{10(3n-1)\sigma _{3}(n)+\sigma _{1}(n)+240\sum _{0<k<n}\sigma _{1}(k)\sigma _{3}(n-k)\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef679c37f53505899efb2f332faf2b9c22ffef8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:69.081ex; height:7.509ex;" alt="{\displaystyle \sigma _{5}(n)={\frac {1}{21}}\left\{10(3n-1)\sigma _{3}(n)+\sigma _{1}(n)+240\sum _{0<k<n}\sigma _{1}(k)\sigma _{3}(n-k)\right\}.}"></span>    <sup id="cite_ref-Koblitz,_ex._III.2.8_30-0" class="reference"><a href="#cite_note-Koblitz,_ex._III.2.8-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sigma _{7}(n)&={\frac {1}{20}}\left\{21(2n-1)\sigma _{5}(n)-\sigma _{1}(n)+504\sum _{0<k<n}\sigma _{1}(k)\sigma _{5}(n-k)\right\}\\&=\sigma _{3}(n)+120\sum _{0<k<n}\sigma _{3}(k)\sigma _{3}(n-k).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>20</mn> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow> <mn>21</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>504</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>120</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{7}(n)&={\frac {1}{20}}\left\{21(2n-1)\sigma _{5}(n)-\sigma _{1}(n)+504\sum _{0<k<n}\sigma _{1}(k)\sigma _{5}(n-k)\right\}\\&=\sigma _{3}(n)+120\sum _{0<k<n}\sigma _{3}(k)\sigma _{3}(n-k).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60804e1d6ff542e7aa015bfedd88cad618e4dbb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:68.799ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\sigma _{7}(n)&={\frac {1}{20}}\left\{21(2n-1)\sigma _{5}(n)-\sigma _{1}(n)+504\sum _{0<k<n}\sigma _{1}(k)\sigma _{5}(n-k)\right\}\\&=\sigma _{3}(n)+120\sum _{0<k<n}\sigma _{3}(k)\sigma _{3}(n-k).\end{aligned}}}"></span>    <sup id="cite_ref-Koblitz,_ex._III.2.8_30-1" class="reference"><a href="#cite_note-Koblitz,_ex._III.2.8-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sigma _{9}(n)&={\frac {1}{11}}\left\{10(3n-2)\sigma _{7}(n)+\sigma _{1}(n)+480\sum _{0<k<n}\sigma _{1}(k)\sigma _{7}(n-k)\right\}\\&={\frac {1}{11}}\left\{21\sigma _{5}(n)-10\sigma _{3}(n)+5040\sum _{0<k<n}\sigma _{3}(k)\sigma _{5}(n-k)\right\}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>11</mn> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow> <mn>10</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>480</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>11</mn> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow> <mn>21</mn> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>10</mn> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>5040</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{9}(n)&={\frac {1}{11}}\left\{10(3n-2)\sigma _{7}(n)+\sigma _{1}(n)+480\sum _{0<k<n}\sigma _{1}(k)\sigma _{7}(n-k)\right\}\\&={\frac {1}{11}}\left\{21\sigma _{5}(n)-10\sigma _{3}(n)+5040\sum _{0<k<n}\sigma _{3}(k)\sigma _{5}(n-k)\right\}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba426096859105e768bfe3356f779f57518fce15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:68.799ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}\sigma _{9}(n)&={\frac {1}{11}}\left\{10(3n-2)\sigma _{7}(n)+\sigma _{1}(n)+480\sum _{0<k<n}\sigma _{1}(k)\sigma _{7}(n-k)\right\}\\&={\frac {1}{11}}\left\{21\sigma _{5}(n)-10\sigma _{3}(n)+5040\sum _{0<k<n}\sigma _{3}(k)\sigma _{5}(n-k)\right\}.\end{aligned}}}"></span>    <sup id="cite_ref-Ramanujan,_p._146_29-1" class="reference"><a href="#cite_note-Ramanujan,_p._146-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau (n)={\frac {65}{756}}\sigma _{11}(n)+{\frac {691}{756}}\sigma _{5}(n)-{\frac {691}{3}}\sum _{0<k<n}\sigma _{5}(k)\sigma _{5}(n-k),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>65</mn> <mn>756</mn> </mfrac> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>691</mn> <mn>756</mn> </mfrac> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>691</mn> <mn>3</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau (n)={\frac {65}{756}}\sigma _{11}(n)+{\frac {691}{756}}\sigma _{5}(n)-{\frac {691}{3}}\sum _{0<k<n}\sigma _{5}(k)\sigma _{5}(n-k),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63f715859ad8220fb82516e61358dece84f74c22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.831ex; height:6.509ex;" alt="{\displaystyle \tau (n)={\frac {65}{756}}\sigma _{11}(n)+{\frac {691}{756}}\sigma _{5}(n)-{\frac {691}{3}}\sum _{0<k<n}\sigma _{5}(k)\sigma _{5}(n-k),}"></span>     where <i>τ</i>(<i>n</i>) is Ramanujan's function.    <sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Since <i>σ</i><sub><i>k</i></sub>(<i>n</i>) (for natural number <i>k</i>) and <i>τ</i>(<i>n</i>) are integers, the above formulas can be used to prove congruences<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> for the functions. See <a href="/wiki/Ramanujan_tau_function" title="Ramanujan tau function">Ramanujan tau function</a> for some examples. </p><p>Extend the domain of the partition function by setting <span class="texhtml"><i>p</i>(0) = 1.</span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(n)={\frac {1}{n}}\sum _{1\leq k\leq n}\sigma (k)p(n-k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(n)={\frac {1}{n}}\sum _{1\leq k\leq n}\sigma (k)p(n-k).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/669758ab94018ba29ad8c3e4c6f1d2ba757bd8d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.089ex; width:29.21ex; height:6.676ex;" alt="{\displaystyle p(n)={\frac {1}{n}}\sum _{1\leq k\leq n}\sigma (k)p(n-k).}"></span>    <sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup>   This recurrence can be used to compute <i>p</i>(<i>n</i>).</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Class_number_related">Class number related</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=35" title="Edit section: Class number related"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Lejeune Dirichlet</a> discovered formulas that relate the class number <i>h</i> of <a href="/wiki/Quadratic_number_field" class="mw-redirect" title="Quadratic number field">quadratic number fields</a> to the Jacobi symbol.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p><p>An integer <i>D</i> is called a <b>fundamental discriminant</b> if it is the <a href="/wiki/Discriminant" title="Discriminant">discriminant</a> of a quadratic number field. This is equivalent to <i>D</i> ≠ 1 and either a) <i>D</i> is <a href="/wiki/Squarefree" class="mw-redirect" title="Squarefree">squarefree</a> and <i>D</i> ≡ 1 (mod 4) or b) <i>D</i> ≡ 0 (mod 4), <i>D</i>/4 is squarefree, and <i>D</i>/4 ≡ 2 or 3 (mod 4).<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p><p>Extend the Jacobi symbol to accept even numbers in the "denominator" by defining the <a href="/wiki/Kronecker_symbol" title="Kronecker symbol">Kronecker symbol</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {a}{2}}\right)={\begin{cases}\;\;\,0&{\text{ if }}a{\text{ is even}}\\(-1)^{\frac {a^{2}-1}{8}}&{\text{ if }}a{\text{ is odd. }}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is even</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mn>8</mn> </mfrac> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>a</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 \left({\frac {a}{2}}\right)={\begin{cases}\;\;\,0&{\text{ if }}a{\text{ is even}}\\(-1)^{\frac {a^{2}-1}{8}}&{\text{ if }}a{\text{ is odd. }}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/572fda57e8d9fc0d6206daa983433efa9324317f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.397ex; height:7.509ex;" alt="{\displaystyle \left({\frac {a}{2}}\right)={\begin{cases}\;\;\,0&{\text{ if }}a{\text{ is even}}\\(-1)^{\frac {a^{2}-1}{8}}&{\text{ if }}a{\text{ is odd. }}\end{cases}}}"></span> </p><p>Then if <i>D</i> < −4 is a fundamental discriminant<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}h(D)&={\frac {1}{D}}\sum _{r=1}^{|D|}r\left({\frac {D}{r}}\right)\\&={\frac {1}{2-\left({\tfrac {D}{2}}\right)}}\sum _{r=1}^{|D|/2}\left({\frac {D}{r}}\right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>h</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>D</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </munderover> <mi>r</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>D</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h(D)&={\frac {1}{D}}\sum _{r=1}^{|D|}r\left({\frac {D}{r}}\right)\\&={\frac {1}{2-\left({\tfrac {D}{2}}\right)}}\sum _{r=1}^{|D|/2}\left({\frac {D}{r}}\right).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e53c791d5cfcd63a6ce04f49d475a3962dbb3b9b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.617ex; margin-bottom: -0.221ex; width:30.643ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}h(D)&={\frac {1}{D}}\sum _{r=1}^{|D|}r\left({\frac {D}{r}}\right)\\&={\frac {1}{2-\left({\tfrac {D}{2}}\right)}}\sum _{r=1}^{|D|/2}\left({\frac {D}{r}}\right).\end{aligned}}}"></span> </p><p>There is also a formula relating <i>r</i><sub>3</sub> and <i>h</i>. Again, let <i>D</i> be a fundamental discriminant, <i>D</i> < −4. Then<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}(|D|)=12\left(1-\left({\frac {D}{2}}\right)\right)h(D).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>12</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}(|D|)=12\left(1-\left({\frac {D}{2}}\right)\right)h(D).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ed7980bf45f0bbf2b0a3ca2e697e8c96ebd77a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.653ex; height:6.176ex;" alt="{\displaystyle r_{3}(|D|)=12\left(1-\left({\frac {D}{2}}\right)\right)h(D).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Prime-count_related">Prime-count related</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=36" title="Edit section: Prime-count related"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be2447b1119dfedd79acffdaf38c884e041cc4e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.723ex; height:5.176ex;" alt="{\displaystyle H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}}"></span>   be the <i>n</i>th <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic number</a>. Then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (n)\leq H_{n}+e^{H_{n}}\log H_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mi>log</mi> <mo>⁡</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (n)\leq H_{n}+e^{H_{n}}\log H_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46213fe5a0a864edea08701f6685418e9a7ca63c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.164ex; height:3.176ex;" alt="{\displaystyle \sigma (n)\leq H_{n}+e^{H_{n}}\log H_{n}}"></span>   is true for every natural number <i>n</i> if and only if the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> is true.    <sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup></dd></dl> <p>The Riemann hypothesis is also equivalent to the statement that, for all <i>n</i> > 5040, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (n)<e^{\gamma }n\log \log n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo><</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (n)<e^{\gamma }n\log \log n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e169504a533384cc31b217ecb6749fc634a22fd7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.735ex; height:2.843ex;" alt="{\displaystyle \sigma (n)<e^{\gamma }n\log \log n}"></span> (where γ is the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>). This is <a href="/wiki/Divisor_function#Growth_rate" title="Divisor function">Robin's theorem</a>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{p}\nu _{p}(n)=\Omega (n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munder> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{p}\nu _{p}(n)=\Omega (n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f0520c26d28903f0b2e17dac96ba768ee3c5a9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.781ex; height:5.676ex;" alt="{\displaystyle \sum _{p}\nu _{p}(n)=\Omega (n).}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)=\sum _{n\leq x}\Lambda (n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)=\sum _{n\leq x}\Lambda (n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b9f73f28b91d91af2f0f8e927d24be3160d4f1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.957ex; height:5.676ex;" alt="{\displaystyle \psi (x)=\sum _{n\leq x}\Lambda (n).}"></span>    <sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi (x)=\sum _{n\leq x}{\frac {\Lambda (n)}{\log n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi (x)=\sum _{n\leq x}{\frac {\Lambda (n)}{\log n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f09f36af683ce08ee290af00b923f4c2a74020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.023ex; height:7.009ex;" alt="{\displaystyle \Pi (x)=\sum _{n\leq x}{\frac {\Lambda (n)}{\log n}}.}"></span>    <sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\theta (x)}=\prod _{p\leq x}p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{\theta (x)}=\prod _{p\leq x}p.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9cd8ef5619e6cbe18c811c9e67592f35841c798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:12.654ex; height:5.843ex;" alt="{\displaystyle e^{\theta (x)}=\prod _{p\leq x}p.}"></span>    <sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\psi (x)}=\operatorname {lcm} [1,2,\dots ,\lfloor x\rfloor ].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>lcm</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{\psi (x)}=\operatorname {lcm} [1,2,\dots ,\lfloor x\rfloor ].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f0671a33f6f269f9f767e7b72dc0383a7fc7c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.191ex; height:3.343ex;" alt="{\displaystyle e^{\psi (x)}=\operatorname {lcm} [1,2,\dots ,\lfloor x\rfloor ].}"></span>    <sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Menon's_identity"><span id="Menon.27s_identity"></span>Menon's identity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=37" title="Edit section: Menon's identity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1965 <a href="/wiki/P_Kesava_Menon" title="P Kesava Menon">P Kesava Menon</a> proved<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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> <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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a43aec65017d2b1bbea12a7f99b4e1f1275d7b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:34.362ex; height:7.676ex;" alt="{\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\gcd(k-1,n)=\varphi (n)d(n).}"></span> </p><p>This has been generalized by a number of mathematicians. For example, </p> <ul><li>B. Sury<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\stackrel {1\leq k_{1},k_{2},\dots ,k_{s}\leq n}{\gcd(k_{1},n)=1}}\gcd(k_{1}-1,k_{2},\dots ,k_{s},n)=\varphi (n)\sigma _{s-1}(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> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <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> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>≤</mo> <mi>n</mi> </mrow> </mover> </mrow> </mrow> </munder> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <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> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <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_{1},k_{2},\dots ,k_{s}\leq n}{\gcd(k_{1},n)=1}}\gcd(k_{1}-1,k_{2},\dots ,k_{s},n)=\varphi (n)\sigma _{s-1}(n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9c9206305a5e993aaad9d5eba88e57203b65e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:51.846ex; height:7.843ex;" alt="{\displaystyle \sum _{\stackrel {1\leq k_{1},k_{2},\dots ,k_{s}\leq n}{\gcd(k_{1},n)=1}}\gcd(k_{1}-1,k_{2},\dots ,k_{s},n)=\varphi (n)\sigma _{s-1}(n).}"></span></li> <li>N. Rao<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\stackrel {1\leq k_{1},k_{2},\dots ,k_{s}\leq n}{\gcd(k_{1},k_{2},\dots ,k_{s},n)=1}}\gcd(k_{1}-a_{1},k_{2}-a_{2},\dots ,k_{s}-a_{s},n)^{s}=J_{s}(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> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <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> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>≤</mo> <mi>n</mi> </mrow> </mover> </mrow> </mrow> </munder> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>,</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <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_{1},k_{2},\dots ,k_{s}\leq n}{\gcd(k_{1},k_{2},\dots ,k_{s},n)=1}}\gcd(k_{1}-a_{1},k_{2}-a_{2},\dots ,k_{s}-a_{s},n)^{s}=J_{s}(n)d(n),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f982b6eb17f4c6c8c7b628b45a86172e7f7ab0b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:66.669ex; height:7.843ex;" alt="{\displaystyle \sum _{\stackrel {1\leq k_{1},k_{2},\dots ,k_{s}\leq n}{\gcd(k_{1},k_{2},\dots ,k_{s},n)=1}}\gcd(k_{1}-a_{1},k_{2}-a_{2},\dots ,k_{s}-a_{s},n)^{s}=J_{s}(n)d(n),}"></span> where <i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, ..., <i>a</i><sub><i>s</i></sub> are integers, gcd(<i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, ..., <i>a</i><sub><i>s</i></sub>, <i>n</i>) = 1.</li> <li><a href="/wiki/L%C3%A1szl%C3%B3_Fejes_T%C3%B3th" title="László Fejes Tóth">László Fejes Tóth</a><sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\stackrel {1\leq k\leq m}{\gcd(k,m)=1}}\gcd(k^{2}-1,m_{1})\gcd(k^{2}-1,m_{2})=\varphi (n)\sum _{\stackrel {d_{1}\mid m_{1}}{d_{2}\mid m_{2}}}\varphi (\gcd(d_{1},d_{2}))2^{\omega (\operatorname {lcm} (d_{1},d_{2}))},}"> <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>m</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>m</mi> </mrow> </mover> </mrow> </mrow> </munder> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∣</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∣</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mover> </mrow> </mrow> </munder> <mi>φ</mi> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>lcm</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\stackrel {1\leq k\leq m}{\gcd(k,m)=1}}\gcd(k^{2}-1,m_{1})\gcd(k^{2}-1,m_{2})=\varphi (n)\sum _{\stackrel {d_{1}\mid m_{1}}{d_{2}\mid m_{2}}}\varphi (\gcd(d_{1},d_{2}))2^{\omega (\operatorname {lcm} (d_{1},d_{2}))},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41de1e8b3191a70ce78abb0171761ec92edf5314" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:79.974ex; height:7.843ex;" alt="{\displaystyle \sum _{\stackrel {1\leq k\leq m}{\gcd(k,m)=1}}\gcd(k^{2}-1,m_{1})\gcd(k^{2}-1,m_{2})=\varphi (n)\sum _{\stackrel {d_{1}\mid m_{1}}{d_{2}\mid m_{2}}}\varphi (\gcd(d_{1},d_{2}))2^{\omega (\operatorname {lcm} (d_{1},d_{2}))},}"></span> where <i>m</i><sub>1</sub> and <i>m</i><sub>2</sub> are odd, <i>m</i> = lcm(<i>m</i><sub>1</sub>, <i>m</i><sub>2</sub>).</li></ul> <p>In fact, if <i>f</i> is any arithmetical function<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}f(\gcd(k-1,n))=\varphi (n)\sum _{d\mid n}{\frac {(\mu *f)(d)}{\varphi (d)}},}"> <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> <mi>f</mi> <mo stretchy="false">(</mo> <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 stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <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> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>∗</mo> <mi>f</mi> <mo stretchy="false">)</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}f(\gcd(k-1,n))=\varphi (n)\sum _{d\mid n}{\frac {(\mu *f)(d)}{\varphi (d)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66f7efffdcae2164e6d4073e753d003aa9c11adc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:47.704ex; height:9.009ex;" alt="{\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}f(\gcd(k-1,n))=\varphi (n)\sum _{d\mid n}{\frac {(\mu *f)(d)}{\varphi (d)}},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}"></span> stands for Dirichlet convolution. </p> <div class="mw-heading mw-heading3"><h3 id="Miscellaneous">Miscellaneous</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=38" title="Edit section: Miscellaneous"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>m</i> and <i>n</i> be distinct, odd, and positive. Then the Jacobi symbol satisfies the law of <a href="/wiki/Quadratic_reciprocity" title="Quadratic reciprocity">quadratic reciprocity</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {m}{n}}\right)\left({\frac {n}{m}}\right)=(-1)^{(m-1)(n-1)/4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>m</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {m}{n}}\right)\left({\frac {n}{m}}\right)=(-1)^{(m-1)(n-1)/4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1859f7d2ae98869a471b5c38dd0881ece13ab9bf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.281ex; height:4.843ex;" alt="{\displaystyle \left({\frac {m}{n}}\right)\left({\frac {n}{m}}\right)=(-1)^{(m-1)(n-1)/4}.}"></span> </p><p>Let <i>D</i>(<i>n</i>) be the arithmetic derivative. Then the logarithmic derivative <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {D(n)}{n}}=\sum _{\stackrel {p\mid n}{p{\text{ prime}}}}{\frac {v_{p}(n)}{p}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> prime</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>∣</mo> <mi>n</mi> </mrow> </mover> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mi>p</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D(n)}{n}}=\sum _{\stackrel {p\mid n}{p{\text{ prime}}}}{\frac {v_{p}(n)}{p}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a93003104e442fef9683c36ab87dfd1ac48aac7f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:21.846ex; height:9.009ex;" alt="{\displaystyle {\frac {D(n)}{n}}=\sum _{\stackrel {p\mid n}{p{\text{ prime}}}}{\frac {v_{p}(n)}{p}}.}"></span> See <a href="/wiki/Arithmetic_derivative" title="Arithmetic derivative">Arithmetic derivative</a> for details. </p><p>Let <i>λ</i>(<i>n</i>) be Liouville's function. Then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\lambda (n)|\mu (n)=\lambda (n)|\mu (n)|=\mu (n),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\lambda (n)|\mu (n)=\lambda (n)|\mu (n)|=\mu (n),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e83d50f3e87c51707740040fe9988b5d7aace06e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.367ex; height:2.843ex;" alt="{\displaystyle |\lambda (n)|\mu (n)=\lambda (n)|\mu (n)|=\mu (n),}"></span>     and</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (n)\mu (n)=|\mu (n)|=\mu ^{2}(n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda (n)\mu (n)=|\mu (n)|=\mu ^{2}(n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84426eeb26447a1ec2ab04401ae74a5ae82ff22b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.568ex; height:3.176ex;" alt="{\displaystyle \lambda (n)\mu (n)=|\mu (n)|=\mu ^{2}(n).}"></span>    </dd></dl> <p>Let <i>λ</i>(<i>n</i>) be Carmichael's function. Then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (n)\mid \phi (n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda (n)\mid \phi (n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f9778d573cb65f9f60016068a0ac0ea9ccb6e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.733ex; height:2.843ex;" alt="{\displaystyle \lambda (n)\mid \phi (n).}"></span>     Further,</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (n)=\phi (n){\text{ if and only if }}n={\begin{cases}1,2,4;\\3,5,7,9,11,\ldots {\text{ (that is, }}p^{k}{\text{, where }}p{\text{ is an odd prime)}};\\6,10,14,18,\ldots {\text{ (that is, }}2p^{k}{\text{, where }}p{\text{ is an odd prime)}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> if and only if </mtext> </mrow> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>9</mn> <mo>,</mo> <mn>11</mn> <mo>,</mo> <mo>…</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> (that is, </mtext> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>, where </mtext> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is an odd prime)</mtext> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> <mo>,</mo> <mn>10</mn> <mo>,</mo> <mn>14</mn> <mo>,</mo> <mn>18</mn> <mo>,</mo> <mo>…</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> (that is, </mtext> </mrow> <mn>2</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>, where </mtext> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is an odd prime)</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda (n)=\phi (n){\text{ if and only if }}n={\begin{cases}1,2,4;\\3,5,7,9,11,\ldots {\text{ (that is, }}p^{k}{\text{, where }}p{\text{ is an odd prime)}};\\6,10,14,18,\ldots {\text{ (that is, }}2p^{k}{\text{, where }}p{\text{ is an odd prime)}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1700918ad857c4e5f2a7eb54445e2785e50e5bcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:87.563ex; height:8.843ex;" alt="{\displaystyle \lambda (n)=\phi (n){\text{ if and only if }}n={\begin{cases}1,2,4;\\3,5,7,9,11,\ldots {\text{ (that is, }}p^{k}{\text{, where }}p{\text{ is an odd prime)}};\\6,10,14,18,\ldots {\text{ (that is, }}2p^{k}{\text{, where }}p{\text{ is an odd prime)}}.\end{cases}}}"></span></dd></dl> <p>See <a href="/wiki/Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">Multiplicative group of integers modulo n</a> and <a href="/wiki/Primitive_root_modulo_n" title="Primitive root modulo n">Primitive root modulo n</a>.   </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\omega (n)}\leq d(n)\leq 2^{\Omega (n)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\omega (n)}\leq d(n)\leq 2^{\Omega (n)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f2ddab8590b6372bfdb32df3dc74545c872223" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.793ex; height:3.343ex;" alt="{\displaystyle 2^{\omega (n)}\leq d(n)\leq 2^{\Omega (n)}.}"></span>    <sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {6}{\pi ^{2}}}<{\frac {\phi (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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {6}{\pi ^{2}}}<{\frac {\phi (n)\sigma (n)}{n^{2}}}<1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/557acffb6f82d40cae6ae4369f2c63769e4bb8ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.19ex; height:6.009ex;" alt="{\displaystyle {\frac {6}{\pi ^{2}}}<{\frac {\phi (n)\sigma (n)}{n^{2}}}<1.}"></span>    <sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}c_{q}(n)&={\frac {\mu \left({\frac {q}{\gcd(q,n)}}\right)}{\phi \left({\frac {q}{\gcd(q,n)}}\right)}}\phi (q)\\&=\sum _{\delta \mid \gcd(q,n)}\mu \left({\frac {q}{\delta }}\right)\delta .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mrow> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>ϕ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mrow> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mi>μ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mi>δ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c_{q}(n)&={\frac {\mu \left({\frac {q}{\gcd(q,n)}}\right)}{\phi \left({\frac {q}{\gcd(q,n)}}\right)}}\phi (q)\\&=\sum _{\delta \mid \gcd(q,n)}\mu \left({\frac {q}{\delta }}\right)\delta .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4c2fd999f96339479e40edc3e5421374f8aa19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:25.495ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}c_{q}(n)&={\frac {\mu \left({\frac {q}{\gcd(q,n)}}\right)}{\phi \left({\frac {q}{\gcd(q,n)}}\right)}}\phi (q)\\&=\sum _{\delta \mid \gcd(q,n)}\mu \left({\frac {q}{\delta }}\right)\delta .\end{aligned}}}"></span>    <sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup>     Note that  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (q)=\sum _{\delta \mid q}\mu \left({\frac {q}{\delta }}\right)\delta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>q</mi> </mrow> </munder> <mi>μ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mi>δ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (q)=\sum _{\delta \mid q}\mu \left({\frac {q}{\delta }}\right)\delta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4856244f844fd39211f9653b7a9bb77c4adc0ecb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:19.657ex; height:6.509ex;" alt="{\displaystyle \phi (q)=\sum _{\delta \mid q}\mu \left({\frac {q}{\delta }}\right)\delta .}"></span>    <sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{q}(1)=\mu (q).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{q}(1)=\mu (q).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1baad8c1e5bc91763239fc1282a55ed73e8b324f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.993ex; height:3.009ex;" alt="{\displaystyle c_{q}(1)=\mu (q).}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{q}(q)=\phi (q).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{q}(q)=\phi (q).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/999756644a94b7440e6d2eb3ee47b4b8cf69769f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.884ex; height:3.009ex;" alt="{\displaystyle c_{q}(q)=\phi (q).}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\delta \mid n}d^{3}(\delta )=\left(\sum _{\delta \mid n}d(\delta )\right)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\delta \mid n}d^{3}(\delta )=\left(\sum _{\delta \mid n}d(\delta )\right)^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615395bc95821e3397f18af1020b02377b33b552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.555ex; height:8.676ex;" alt="{\displaystyle \sum _{\delta \mid n}d^{3}(\delta )=\left(\sum _{\delta \mid n}d(\delta )\right)^{2}.}"></span>    <sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup>   Compare this with <span class="texhtml">1<sup>3</sup> + 2<sup>3</sup> + 3<sup>3</sup> + ... + <i>n</i><sup>3</sup> = (1 + 2 + 3 + ... + <i>n</i>)<sup>2</sup></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(uv)=\sum _{\delta \mid \gcd(u,v)}\mu (\delta )d\left({\frac {u}{\delta }}\right)d\left({\frac {v}{\delta }}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>u</mi> <mi>δ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>δ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(uv)=\sum _{\delta \mid \gcd(u,v)}\mu (\delta )d\left({\frac {u}{\delta }}\right)d\left({\frac {v}{\delta }}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e81ad86fba6b2eac770597516167249f373d22ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:34.674ex; height:6.509ex;" alt="{\displaystyle d(uv)=\sum _{\delta \mid \gcd(u,v)}\mu (\delta )d\left({\frac {u}{\delta }}\right)d\left({\frac {v}{\delta }}\right).}"></span>    <sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{k}(u)\sigma _{k}(v)=\sum _{\delta \mid \gcd(u,v)}\delta ^{k}\sigma _{k}\left({\frac {uv}{\delta ^{2}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </munder> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mi>v</mi> </mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{k}(u)\sigma _{k}(v)=\sum _{\delta \mid \gcd(u,v)}\delta ^{k}\sigma _{k}\left({\frac {uv}{\delta ^{2}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63ed3429afd4bc31d042c4f6a62bab8d04fe19cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:34.227ex; height:7.176ex;" alt="{\displaystyle \sigma _{k}(u)\sigma _{k}(v)=\sum _{\delta \mid \gcd(u,v)}\delta ^{k}\sigma _{k}\left({\frac {uv}{\delta ^{2}}}\right).}"></span>    <sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau (u)\tau (v)=\sum _{\delta \mid \gcd(u,v)}\delta ^{11}\tau \left({\frac {uv}{\delta ^{2}}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>∣</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </munder> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msup> <mi>τ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mi>v</mi> </mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau (u)\tau (v)=\sum _{\delta \mid \gcd(u,v)}\delta ^{11}\tau \left({\frac {uv}{\delta ^{2}}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe65c15e16eb3d465e012cc7678a014b15fba9b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:31.372ex; height:7.176ex;" alt="{\displaystyle \tau (u)\tau (v)=\sum _{\delta \mid \gcd(u,v)}\delta ^{11}\tau \left({\frac {uv}{\delta ^{2}}}\right),}"></span>     where <i>τ</i>(<i>n</i>) is Ramanujan's function.    <sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading2"><h2 id="First_100_values_of_some_arithmetic_functions">First 100 values of some arithmetic functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=39" title="Edit section: First 100 values of some arithmetic functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" style="text-align:right;"> <tbody><tr> <th><i>n</i></th> <th>factorization</th> <th>𝜙(<i>n</i>)</th> <th><i>ω</i>(<i>n</i>)</th> <th>Ω(<i>n</i>)</th> <th>𝜆(<i>n</i>)</th> <th>𝜇(<i>n</i>)</th> <th>𝛬(<i>n</i>)</th> <th><span class="texhtml mvar" style="font-style:italic;">π</span>(<i>n</i>)</th> <th>𝜎<sub>0</sub>(<i>n</i>)</th> <th>𝜎<sub>1</sub>(<i>n</i>)</th> <th>𝜎<sub>2</sub>(<i>n</i>)</th> <th><i>r</i><sub>2</sub>(<i>n</i>)</th> <th><i>r</i><sub>3</sub>(<i>n</i>)</th> <th><i>r</i><sub>4</sub>(<i>n</i>) </th></tr> <tr> <td>1</td> <td style="text-align:center;">1</td> <td>1</td> <td>0</td> <td>0</td> <td>1</td> <td>1</td> <td>0</td> <td>0</td> <td>1</td> <td>1</td> <td>1</td> <td>4</td> <td>6</td> <td>8 </td></tr> <tr style="background-color:#ddeeff;"> <td>2</td> <td style="text-align:center;">2</td> <td>1</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>0.69</td> <td>1</td> <td>2</td> <td>3</td> <td>5</td> <td>4</td> <td>12</td> <td>24 </td></tr> <tr style="background-color:#ddeeff;"> <td>3</td> <td style="text-align:center;">3</td> <td>2</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>1.10</td> <td>2</td> <td>2</td> <td>4</td> <td>10</td> <td>0</td> <td>8</td> <td>32 </td></tr> <tr> <td>4</td> <td style="text-align:center;">2<sup>2</sup></td> <td>2</td> <td>1</td> <td>2</td> <td>1</td> <td>0</td> <td>0.69</td> <td>2</td> <td>3</td> <td>7</td> <td>21</td> <td>4</td> <td>6</td> <td>24 </td></tr> <tr style="background-color:#ddeeff;"> <td>5</td> <td style="text-align:center;">5</td> <td>4</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>1.61</td> <td>3</td> <td>2</td> <td>6</td> <td>26</td> <td>8</td> <td>24</td> <td>48 </td></tr> <tr> <td>6</td> <td style="text-align:center;">2 · 3</td> <td>2</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>3</td> <td>4</td> <td>12</td> <td>50</td> <td>0</td> <td>24</td> <td>96 </td></tr> <tr style="background-color:#ddeeff;"> <td>7</td> <td style="text-align:center;">7</td> <td>6</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>1.95</td> <td>4</td> <td>2</td> <td>8</td> <td>50</td> <td>0</td> <td>0</td> <td>64 </td></tr> <tr> <td>8</td> <td style="text-align:center;">2<sup>3</sup></td> <td>4</td> <td>1</td> <td>3</td> <td>−1</td> <td>0</td> <td>0.69</td> <td>4</td> <td>4</td> <td>15</td> <td>85</td> <td>4</td> <td>12</td> <td>24 </td></tr> <tr> <td>9</td> <td style="text-align:center;">3<sup>2</sup></td> <td>6</td> <td>1</td> <td>2</td> <td>1</td> <td>0</td> <td>1.10</td> <td>4</td> <td>3</td> <td>13</td> <td>91</td> <td>4</td> <td>30</td> <td>104 </td></tr> <tr> <td>10</td> <td style="text-align:center;">2 · 5</td> <td>4</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>4</td> <td>4</td> <td>18</td> <td>130</td> <td>8</td> <td>24</td> <td>144 </td></tr> <tr style="background-color:#ddeeff;"> <td>11</td> <td style="text-align:center;">11</td> <td>10</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>2.40</td> <td>5</td> <td>2</td> <td>12</td> <td>122</td> <td>0</td> <td>24</td> <td>96 </td></tr> <tr> <td>12</td> <td style="text-align:center;">2<sup>2</sup> · 3</td> <td>4</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>5</td> <td>6</td> <td>28</td> <td>210</td> <td>0</td> <td>8</td> <td>96 </td></tr> <tr style="background-color:#ddeeff;"> <td>13</td> <td style="text-align:center;">13</td> <td>12</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>2.56</td> <td>6</td> <td>2</td> <td>14</td> <td>170</td> <td>8</td> <td>24</td> <td>112 </td></tr> <tr> <td>14</td> <td style="text-align:center;">2 · 7</td> <td>6</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>6</td> <td>4</td> <td>24</td> <td>250</td> <td>0</td> <td>48</td> <td>192 </td></tr> <tr> <td>15</td> <td style="text-align:center;">3 · 5</td> <td>8</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>6</td> <td>4</td> <td>24</td> <td>260</td> <td>0</td> <td>0</td> <td>192 </td></tr> <tr> <td>16</td> <td style="text-align:center;">2<sup>4</sup></td> <td>8</td> <td>1</td> <td>4</td> <td>1</td> <td>0</td> <td>0.69</td> <td>6</td> <td>5</td> <td>31</td> <td>341</td> <td>4</td> <td>6</td> <td>24 </td></tr> <tr style="background-color:#ddeeff;"> <td>17</td> <td style="text-align:center;">17</td> <td>16</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>2.83</td> <td>7</td> <td>2</td> <td>18</td> <td>290</td> <td>8</td> <td>48</td> <td>144 </td></tr> <tr> <td>18</td> <td style="text-align:center;">2 · 3<sup>2</sup></td> <td>6</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>7</td> <td>6</td> <td>39</td> <td>455</td> <td>4</td> <td>36</td> <td>312 </td></tr> <tr style="background-color:#ddeeff;"> <td>19</td> <td style="text-align:center;">19</td> <td>18</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>2.94</td> <td>8</td> <td>2</td> <td>20</td> <td>362</td> <td>0</td> <td>24</td> <td>160 </td></tr> <tr> <td>20</td> <td style="text-align:center;">2<sup>2</sup> · 5</td> <td>8</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>8</td> <td>6</td> <td>42</td> <td>546</td> <td>8</td> <td>24</td> <td>144 </td></tr> <tr> <td>21</td> <td style="text-align:center;">3 · 7</td> <td>12</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>8</td> <td>4</td> <td>32</td> <td>500</td> <td>0</td> <td>48</td> <td>256 </td></tr> <tr> <td>22</td> <td style="text-align:center;">2 · 11</td> <td>10</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>8</td> <td>4</td> <td>36</td> <td>610</td> <td>0</td> <td>24</td> <td>288 </td></tr> <tr style="background-color:#ddeeff;"> <td>23</td> <td style="text-align:center;">23</td> <td>22</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>3.14</td> <td>9</td> <td>2</td> <td>24</td> <td>530</td> <td>0</td> <td>0</td> <td>192 </td></tr> <tr> <td>24</td> <td style="text-align:center;">2<sup>3</sup> · 3</td> <td>8</td> <td>2</td> <td>4</td> <td>1</td> <td>0</td> <td>0</td> <td>9</td> <td>8</td> <td>60</td> <td>850</td> <td>0</td> <td>24</td> <td>96 </td></tr> <tr> <td>25</td> <td style="text-align:center;">5<sup>2</sup></td> <td>20</td> <td>1</td> <td>2</td> <td>1</td> <td>0</td> <td>1.61</td> <td>9</td> <td>3</td> <td>31</td> <td>651</td> <td>12</td> <td>30</td> <td>248 </td></tr> <tr> <td>26</td> <td style="text-align:center;">2 · 13</td> <td>12</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>9</td> <td>4</td> <td>42</td> <td>850</td> <td>8</td> <td>72</td> <td>336 </td></tr> <tr> <td>27</td> <td style="text-align:center;">3<sup>3</sup></td> <td>18</td> <td>1</td> <td>3</td> <td>−1</td> <td>0</td> <td>1.10</td> <td>9</td> <td>4</td> <td>40</td> <td>820</td> <td>0</td> <td>32</td> <td>320 </td></tr> <tr> <td>28</td> <td style="text-align:center;">2<sup>2</sup> · 7</td> <td>12</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>9</td> <td>6</td> <td>56</td> <td>1050</td> <td>0</td> <td>0</td> <td>192 </td></tr> <tr style="background-color:#ddeeff;"> <td>29</td> <td style="text-align:center;">29</td> <td>28</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>3.37</td> <td>10</td> <td>2</td> <td>30</td> <td>842</td> <td>8</td> <td>72</td> <td>240 </td></tr> <tr> <td>30</td> <td style="text-align:center;">2 · 3 · 5</td> <td>8</td> <td>3</td> <td>3</td> <td>−1</td> <td>−1</td> <td>0</td> <td>10</td> <td>8</td> <td>72</td> <td>1300</td> <td>0</td> <td>48</td> <td>576 </td></tr> <tr style="background-color:#ddeeff;"> <td>31</td> <td style="text-align:center;">31</td> <td>30</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>3.43</td> <td>11</td> <td>2</td> <td>32</td> <td>962</td> <td>0</td> <td>0</td> <td>256 </td></tr> <tr> <td>32</td> <td style="text-align:center;">2<sup>5</sup></td> <td>16</td> <td>1</td> <td>5</td> <td>−1</td> <td>0</td> <td>0.69</td> <td>11</td> <td>6</td> <td>63</td> <td>1365</td> <td>4</td> <td>12</td> <td>24 </td></tr> <tr> <td>33</td> <td style="text-align:center;">3 · 11</td> <td>20</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>11</td> <td>4</td> <td>48</td> <td>1220</td> <td>0</td> <td>48</td> <td>384 </td></tr> <tr> <td>34</td> <td style="text-align:center;">2 · 17</td> <td>16</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>11</td> <td>4</td> <td>54</td> <td>1450</td> <td>8</td> <td>48</td> <td>432 </td></tr> <tr> <td>35</td> <td style="text-align:center;">5 · 7</td> <td>24</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>11</td> <td>4</td> <td>48</td> <td>1300</td> <td>0</td> <td>48</td> <td>384 </td></tr> <tr> <td>36</td> <td style="text-align:center;">2<sup>2</sup> · 3<sup>2</sup></td> <td>12</td> <td>2</td> <td>4</td> <td>1</td> <td>0</td> <td>0</td> <td>11</td> <td>9</td> <td>91</td> <td>1911</td> <td>4</td> <td>30</td> <td>312 </td></tr> <tr style="background-color:#ddeeff;"> <td>37</td> <td style="text-align:center;">37</td> <td>36</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>3.61</td> <td>12</td> <td>2</td> <td>38</td> <td>1370</td> <td>8</td> <td>24</td> <td>304 </td></tr> <tr> <td>38</td> <td style="text-align:center;">2 · 19</td> <td>18</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>12</td> <td>4</td> <td>60</td> <td>1810</td> <td>0</td> <td>72</td> <td>480 </td></tr> <tr> <td>39</td> <td style="text-align:center;">3 · 13</td> <td>24</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>12</td> <td>4</td> <td>56</td> <td>1700</td> <td>0</td> <td>0</td> <td>448 </td></tr> <tr> <td>40</td> <td style="text-align:center;">2<sup>3</sup> · 5</td> <td>16</td> <td>2</td> <td>4</td> <td>1</td> <td>0</td> <td>0</td> <td>12</td> <td>8</td> <td>90</td> <td>2210</td> <td>8</td> <td>24</td> <td>144 </td></tr> <tr style="background-color:#ddeeff;"> <td>41</td> <td style="text-align:center;">41</td> <td>40</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>3.71</td> <td>13</td> <td>2</td> <td>42</td> <td>1682</td> <td>8</td> <td>96</td> <td>336 </td></tr> <tr> <td>42</td> <td style="text-align:center;">2 · 3 · 7</td> <td>12</td> <td>3</td> <td>3</td> <td>−1</td> <td>−1</td> <td>0</td> <td>13</td> <td>8</td> <td>96</td> <td>2500</td> <td>0</td> <td>48</td> <td>768 </td></tr> <tr style="background-color:#ddeeff;"> <td>43</td> <td style="text-align:center;">43</td> <td>42</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>3.76</td> <td>14</td> <td>2</td> <td>44</td> <td>1850</td> <td>0</td> <td>24</td> <td>352 </td></tr> <tr> <td>44</td> <td style="text-align:center;">2<sup>2</sup> · 11</td> <td>20</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>14</td> <td>6</td> <td>84</td> <td>2562</td> <td>0</td> <td>24</td> <td>288 </td></tr> <tr> <td>45</td> <td style="text-align:center;">3<sup>2</sup> · 5</td> <td>24</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>14</td> <td>6</td> <td>78</td> <td>2366</td> <td>8</td> <td>72</td> <td>624 </td></tr> <tr> <td>46</td> <td style="text-align:center;">2 · 23</td> <td>22</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>14</td> <td>4</td> <td>72</td> <td>2650</td> <td>0</td> <td>48</td> <td>576 </td></tr> <tr style="background-color:#ddeeff;"> <td>47</td> <td style="text-align:center;">47</td> <td>46</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>3.85</td> <td>15</td> <td>2</td> <td>48</td> <td>2210</td> <td>0</td> <td>0</td> <td>384 </td></tr> <tr> <td>48</td> <td style="text-align:center;">2<sup>4</sup> · 3</td> <td>16</td> <td>2</td> <td>5</td> <td>−1</td> <td>0</td> <td>0</td> <td>15</td> <td>10</td> <td>124</td> <td>3410</td> <td>0</td> <td>8</td> <td>96 </td></tr> <tr> <td>49</td> <td style="text-align:center;">7<sup>2</sup></td> <td>42</td> <td>1</td> <td>2</td> <td>1</td> <td>0</td> <td>1.95</td> <td>15</td> <td>3</td> <td>57</td> <td>2451</td> <td>4</td> <td>54</td> <td>456 </td></tr> <tr> <td>50</td> <td style="text-align:center;">2 · 5<sup>2</sup></td> <td>20</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>15</td> <td>6</td> <td>93</td> <td>3255</td> <td>12</td> <td>84</td> <td>744 </td></tr> <tr> <td>51</td> <td style="text-align:center;">3 · 17</td> <td>32</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>15</td> <td>4</td> <td>72</td> <td>2900</td> <td>0</td> <td>48</td> <td>576 </td></tr> <tr> <td>52</td> <td style="text-align:center;">2<sup>2</sup> · 13</td> <td>24</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>15</td> <td>6</td> <td>98</td> <td>3570</td> <td>8</td> <td>24</td> <td>336 </td></tr> <tr style="background-color:#ddeeff;"> <td>53</td> <td style="text-align:center;">53</td> <td>52</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>3.97</td> <td>16</td> <td>2</td> <td>54</td> <td>2810</td> <td>8</td> <td>72</td> <td>432 </td></tr> <tr> <td>54</td> <td style="text-align:center;">2 · 3<sup>3</sup></td> <td>18</td> <td>2</td> <td>4</td> <td>1</td> <td>0</td> <td>0</td> <td>16</td> <td>8</td> <td>120</td> <td>4100</td> <td>0</td> <td>96</td> <td>960 </td></tr> <tr> <td>55</td> <td style="text-align:center;">5 · 11</td> <td>40</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>16</td> <td>4</td> <td>72</td> <td>3172</td> <td>0</td> <td>0</td> <td>576 </td></tr> <tr> <td>56</td> <td style="text-align:center;">2<sup>3</sup> · 7</td> <td>24</td> <td>2</td> <td>4</td> <td>1</td> <td>0</td> <td>0</td> <td>16</td> <td>8</td> <td>120</td> <td>4250</td> <td>0</td> <td>48</td> <td>192 </td></tr> <tr> <td>57</td> <td style="text-align:center;">3 · 19</td> <td>36</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>16</td> <td>4</td> <td>80</td> <td>3620</td> <td>0</td> <td>48</td> <td>640 </td></tr> <tr> <td>58</td> <td style="text-align:center;">2 · 29</td> <td>28</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>16</td> <td>4</td> <td>90</td> <td>4210</td> <td>8</td> <td>24</td> <td>720 </td></tr> <tr style="background-color:#ddeeff;"> <td>59</td> <td style="text-align:center;">59</td> <td>58</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>4.08</td> <td>17</td> <td>2</td> <td>60</td> <td>3482</td> <td>0</td> <td>72</td> <td>480 </td></tr> <tr> <td>60</td> <td style="text-align:center;">2<sup>2</sup> · 3 · 5</td> <td>16</td> <td>3</td> <td>4</td> <td>1</td> <td>0</td> <td>0</td> <td>17</td> <td>12</td> <td>168</td> <td>5460</td> <td>0</td> <td>0</td> <td>576 </td></tr> <tr style="background-color:#ddeeff;"> <td>61</td> <td style="text-align:center;">61</td> <td>60</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>4.11</td> <td>18</td> <td>2</td> <td>62</td> <td>3722</td> <td>8</td> <td>72</td> <td>496 </td></tr> <tr> <td>62</td> <td style="text-align:center;">2 · 31</td> <td>30</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>18</td> <td>4</td> <td>96</td> <td>4810</td> <td>0</td> <td>96</td> <td>768 </td></tr> <tr> <td>63</td> <td style="text-align:center;">3<sup>2</sup> · 7</td> <td>36</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>18</td> <td>6</td> <td>104</td> <td>4550</td> <td>0</td> <td>0</td> <td>832 </td></tr> <tr> <td>64</td> <td style="text-align:center;">2<sup>6</sup></td> <td>32</td> <td>1</td> <td>6</td> <td>1</td> <td>0</td> <td>0.69</td> <td>18</td> <td>7</td> <td>127</td> <td>5461</td> <td>4</td> <td>6</td> <td>24 </td></tr> <tr> <td>65</td> <td style="text-align:center;">5 · 13</td> <td>48</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>18</td> <td>4</td> <td>84</td> <td>4420</td> <td>16</td> <td>96</td> <td>672 </td></tr> <tr> <td>66</td> <td style="text-align:center;">2 · 3 · 11</td> <td>20</td> <td>3</td> <td>3</td> <td>−1</td> <td>−1</td> <td>0</td> <td>18</td> <td>8</td> <td>144</td> <td>6100</td> <td>0</td> <td>96</td> <td>1152 </td></tr> <tr style="background-color:#ddeeff;"> <td>67</td> <td style="text-align:center;">67</td> <td>66</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>4.20</td> <td>19</td> <td>2</td> <td>68</td> <td>4490</td> <td>0</td> <td>24</td> <td>544 </td></tr> <tr> <td>68</td> <td style="text-align:center;">2<sup>2</sup> · 17</td> <td>32</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>19</td> <td>6</td> <td>126</td> <td>6090</td> <td>8</td> <td>48</td> <td>432 </td></tr> <tr> <td>69</td> <td style="text-align:center;">3 · 23</td> <td>44</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>19</td> <td>4</td> <td>96</td> <td>5300</td> <td>0</td> <td>96</td> <td>768 </td></tr> <tr> <td>70</td> <td style="text-align:center;">2 · 5 · 7</td> <td>24</td> <td>3</td> <td>3</td> <td>−1</td> <td>−1</td> <td>0</td> <td>19</td> <td>8</td> <td>144</td> <td>6500</td> <td>0</td> <td>48</td> <td>1152 </td></tr> <tr style="background-color:#ddeeff;"> <td>71</td> <td style="text-align:center;">71</td> <td>70</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>4.26</td> <td>20</td> <td>2</td> <td>72</td> <td>5042</td> <td>0</td> <td>0</td> <td>576 </td></tr> <tr> <td>72</td> <td style="text-align:center;">2<sup>3</sup> · 3<sup>2</sup></td> <td>24</td> <td>2</td> <td>5</td> <td>−1</td> <td>0</td> <td>0</td> <td>20</td> <td>12</td> <td>195</td> <td>7735</td> <td>4</td> <td>36</td> <td>312 </td></tr> <tr style="background-color:#ddeeff;"> <td>73</td> <td style="text-align:center;">73</td> <td>72</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>4.29</td> <td>21</td> <td>2</td> <td>74</td> <td>5330</td> <td>8</td> <td>48</td> <td>592 </td></tr> <tr> <td>74</td> <td style="text-align:center;">2 · 37</td> <td>36</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>21</td> <td>4</td> <td>114</td> <td>6850</td> <td>8</td> <td>120</td> <td>912 </td></tr> <tr> <td>75</td> <td style="text-align:center;">3 · 5<sup>2</sup></td> <td>40</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>21</td> <td>6</td> <td>124</td> <td>6510</td> <td>0</td> <td>56</td> <td>992 </td></tr> <tr> <td>76</td> <td style="text-align:center;">2<sup>2</sup> · 19</td> <td>36</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>21</td> <td>6</td> <td>140</td> <td>7602</td> <td>0</td> <td>24</td> <td>480 </td></tr> <tr> <td>77</td> <td style="text-align:center;">7 · 11</td> <td>60</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>21</td> <td>4</td> <td>96</td> <td>6100</td> <td>0</td> <td>96</td> <td>768 </td></tr> <tr> <td>78</td> <td style="text-align:center;">2 · 3 · 13</td> <td>24</td> <td>3</td> <td>3</td> <td>−1</td> <td>−1</td> <td>0</td> <td>21</td> <td>8</td> <td>168</td> <td>8500</td> <td>0</td> <td>48</td> <td>1344 </td></tr> <tr style="background-color:#ddeeff;"> <td>79</td> <td style="text-align:center;">79</td> <td>78</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>4.37</td> <td>22</td> <td>2</td> <td>80</td> <td>6242</td> <td>0</td> <td>0</td> <td>640 </td></tr> <tr> <td>80</td> <td style="text-align:center;">2<sup>4</sup> · 5</td> <td>32</td> <td>2</td> <td>5</td> <td>−1</td> <td>0</td> <td>0</td> <td>22</td> <td>10</td> <td>186</td> <td>8866</td> <td>8</td> <td>24</td> <td>144 </td></tr> <tr> <td>81</td> <td style="text-align:center;">3<sup>4</sup></td> <td>54</td> <td>1</td> <td>4</td> <td>1</td> <td>0</td> <td>1.10</td> <td>22</td> <td>5</td> <td>121</td> <td>7381</td> <td>4</td> <td>102</td> <td>968 </td></tr> <tr> <td>82</td> <td style="text-align:center;">2 · 41</td> <td>40</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>22</td> <td>4</td> <td>126</td> <td>8410</td> <td>8</td> <td>48</td> <td>1008 </td></tr> <tr style="background-color:#ddeeff;"> <td>83</td> <td style="text-align:center;">83</td> <td>82</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>4.42</td> <td>23</td> <td>2</td> <td>84</td> <td>6890</td> <td>0</td> <td>72</td> <td>672 </td></tr> <tr> <td>84</td> <td style="text-align:center;">2<sup>2</sup> · 3 · 7</td> <td>24</td> <td>3</td> <td>4</td> <td>1</td> <td>0</td> <td>0</td> <td>23</td> <td>12</td> <td>224</td> <td>10500</td> <td>0</td> <td>48</td> <td>768 </td></tr> <tr> <td>85</td> <td style="text-align:center;">5 · 17</td> <td>64</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>23</td> <td>4</td> <td>108</td> <td>7540</td> <td>16</td> <td>48</td> <td>864 </td></tr> <tr> <td>86</td> <td style="text-align:center;">2 · 43</td> <td>42</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>23</td> <td>4</td> <td>132</td> <td>9250</td> <td>0</td> <td>120</td> <td>1056 </td></tr> <tr> <td>87</td> <td style="text-align:center;">3 · 29</td> <td>56</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>23</td> <td>4</td> <td>120</td> <td>8420</td> <td>0</td> <td>0</td> <td>960 </td></tr> <tr> <td>88</td> <td style="text-align:center;">2<sup>3</sup> · 11</td> <td>40</td> <td>2</td> <td>4</td> <td>1</td> <td>0</td> <td>0</td> <td>23</td> <td>8</td> <td>180</td> <td>10370</td> <td>0</td> <td>24</td> <td>288 </td></tr> <tr style="background-color:#ddeeff;"> <td>89</td> <td style="text-align:center;">89</td> <td>88</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>4.49</td> <td>24</td> <td>2</td> <td>90</td> <td>7922</td> <td>8</td> <td>144</td> <td>720 </td></tr> <tr> <td>90</td> <td style="text-align:center;">2 · 3<sup>2</sup> · 5</td> <td>24</td> <td>3</td> <td>4</td> <td>1</td> <td>0</td> <td>0</td> <td>24</td> <td>12</td> <td>234</td> <td>11830</td> <td>8</td> <td>120</td> <td>1872 </td></tr> <tr> <td>91</td> <td style="text-align:center;">7 · 13</td> <td>72</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>24</td> <td>4</td> <td>112</td> <td>8500</td> <td>0</td> <td>48</td> <td>896 </td></tr> <tr> <td>92</td> <td style="text-align:center;">2<sup>2</sup> · 23</td> <td>44</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>24</td> <td>6</td> <td>168</td> <td>11130</td> <td>0</td> <td>0</td> <td>576 </td></tr> <tr> <td>93</td> <td style="text-align:center;">3 · 31</td> <td>60</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>24</td> <td>4</td> <td>128</td> <td>9620</td> <td>0</td> <td>48</td> <td>1024 </td></tr> <tr> <td>94</td> <td style="text-align:center;">2 · 47</td> <td>46</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>24</td> <td>4</td> <td>144</td> <td>11050</td> <td>0</td> <td>96</td> <td>1152 </td></tr> <tr> <td>95</td> <td style="text-align:center;">5 · 19</td> <td>72</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>24</td> <td>4</td> <td>120</td> <td>9412</td> <td>0</td> <td>0</td> <td>960 </td></tr> <tr> <td>96</td> <td style="text-align:center;">2<sup>5</sup> · 3</td> <td>32</td> <td>2</td> <td>6</td> <td>1</td> <td>0</td> <td>0</td> <td>24</td> <td>12</td> <td>252</td> <td>13650</td> <td>0</td> <td>24</td> <td>96 </td></tr> <tr style="background-color:#ddeeff;"> <td>97</td> <td style="text-align:center;">97</td> <td>96</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>4.57</td> <td>25</td> <td>2</td> <td>98</td> <td>9410</td> <td>8</td> <td>48</td> <td>784 </td></tr> <tr> <td>98</td> <td style="text-align:center;">2 · 7<sup>2</sup></td> <td>42</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>25</td> <td>6</td> <td>171</td> <td>12255</td> <td>4</td> <td>108</td> <td>1368 </td></tr> <tr> <td>99</td> <td style="text-align:center;">3<sup>2</sup> · 11</td> <td>60</td> <td>2</td> <td>3</td> <td>−1</td> <td>0</td> <td>0</td> <td>25</td> <td>6</td> <td>156</td> <td>11102</td> <td>0</td> <td>72</td> <td>1248 </td></tr> <tr> <td>100</td> <td style="text-align:center;">2<sup>2</sup> · 5<sup>2</sup></td> <td>40</td> <td>2</td> <td>4</td> <td>1</td> <td>0</td> <td>0</td> <td>25</td> <td>9</td> <td>217</td> <td>13671</td> <td>12</td> <td>30</td> <td>744 </td></tr> <tr> <th><i>n</i></th> <th>factorization</th> <th>𝜙(<i>n</i>)</th> <th><i>ω</i>(<i>n</i>)</th> <th>Ω(<i>n</i>)</th> <th>𝜆(<i>n</i>)</th> <th>𝜇(<i>n</i>)</th> <th>𝛬(<i>n</i>)</th> <th><span class="texhtml mvar" style="font-style:italic;">π</span>(<i>n</i>)</th> <th>𝜎<sub>0</sub>(<i>n</i>)</th> <th>𝜎<sub>1</sub>(<i>n</i>)</th> <th>𝜎<sub>2</sub>(<i>n</i>)</th> <th><i>r</i><sub>2</sub>(<i>n</i>)</th> <th><i>r</i><sub>3</sub>(<i>n</i>)</th> <th><i>r</i><sub>4</sub>(<i>n</i>) </th></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=40" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFLong1972">Long (1972</a>, p. 151)</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFPettofrezzoByrkit1970">Pettofrezzo & Byrkit (1970</a>, p. 58)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Niven & Zuckerman, 4.2.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Nagell, I.9.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Bateman & Diamond, 2.1.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Hardy & Wright, intro. to Ch. XVI</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Hardy, <i>Ramanujan</i>, § 10.2</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Apostol, <i>Modular Functions ...</i>, § 1.15, Ch. 4, and ch. 6</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Hardy & Wright, §§ 18.1–18.2</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGérald_Tenenbaum1995" class="citation book cs1">Gérald Tenenbaum (1995). <i>Introduction to Analytic and Probabilistic Number Theory</i>. Cambridge studies in advanced mathematics. Vol. 46. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. pp. 36–55. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-41261-7" title="Special:BookSources/0-521-41261-7"><bdi>0-521-41261-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Analytic+and+Probabilistic+Number+Theory&rft.series=Cambridge+studies+in+advanced+mathematics&rft.pages=36-55&rft.pub=Cambridge+University+Press&rft.date=1995&rft.isbn=0-521-41261-7&rft.au=G%C3%A9rald+Tenenbaum&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Hardy & Wright, § 17.6, show how the theory of generating functions can be constructed in a purely formal manner with no attention paid to convergence.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Hardy & Wright, Thm. 263</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Hardy & Wright, Thm. 63</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">see references at <a href="/wiki/Jordan%27s_totient_function" title="Jordan's totient function">Jordan's totient function</a></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Holden et al. in external links The formula is Gegenbauer's</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Hardy & Wright, Thm. 288–290</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Dineva in external links, prop. 4</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Hardy & Wright, Thm. 264</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Hardy & Wright, Thm. 296</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Hardy & Wright, Thm. 278</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Hardy & Wright, Thm. 386</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Hardy, <i>Ramanujan</i>, eqs 9.1.2, 9.1.3</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Koblitz, Ex. III.5.2</span> </li> <li id="cite_note-Hardy_&_Wright,_§_20.13-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hardy_&_Wright,_§_20.13_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hardy_&_Wright,_§_20.13_24-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Hardy & Wright, § 20.13</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">Hardy, <i>Ramanujan</i>, § 9.7</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">Hardy, <i>Ramanujan</i>, § 9.13</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">Hardy, <i>Ramanujan</i>, § 9.17</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">Williams, ch. 13; Huard, et al. (external links).</span> </li> <li id="cite_note-Ramanujan,_p._146-29"><span class="mw-cite-backlink">^ <a href="#cite_ref-Ramanujan,_p._146_29-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Ramanujan,_p._146_29-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Ramanujan, <i>On Certain Arithmetical Functions</i>, Table IV; <i>Papers</i>, p. 146</span> </li> <li id="cite_note-Koblitz,_ex._III.2.8-30"><span class="mw-cite-backlink">^ <a href="#cite_ref-Koblitz,_ex._III.2.8_30-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Koblitz,_ex._III.2.8_30-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Koblitz, ex. III.2.8</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">Koblitz, ex. III.2.3</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">Koblitz, ex. III.2.2</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text">Koblitz, ex. III.2.4</span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Apostol, <i>Modular Functions ...</i>, Ex. 6.10</span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text">Apostol, <i>Modular Functions...</i>, Ch. 6 Ex. 10</span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text">G.H. Hardy, S. Ramannujan, <i>Asymptotic Formulæ in Combinatory Analysis</i>, § 1.3; in Ramannujan, <i>Papers</i> p. 279</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text">Landau, p. 168, credits Gauss as well as Dirichlet</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text">Cohen, Def. 5.1.2</span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text">Cohen, Corr. 5.3.13</span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text">see Edwards, § 9.5 exercises for more complicated formulas.</span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text">Cohen, Prop 5.3.10</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text">See <a href="/wiki/Divisor_function#Growth_rate" title="Divisor function">Divisor function</a>.</span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text">Hardy & Wright, eq. 22.1.2</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text">See <a href="/wiki/Prime-counting_function#Other_prime-counting_functions" title="Prime-counting function">prime-counting functions</a>.</span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text">Hardy & Wright, eq. 22.1.1</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text">Hardy & Wright, eq. 22.1.3</span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text">László Tóth, <i>Menon's Identity and Arithmetical Sums ...</i>, eq. 1</span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text">Tóth, eq. 5</span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text">Tóth, eq. 3</span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text">Tóth, eq. 35</span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text">Tóth, eq. 2</span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text">Tóth states that Menon proved this for multiplicative <i>f</i> in 1965 and V. Sita Ramaiah for general <i>f</i>.</span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text">Hardy <i>Ramanujan</i>, eq. 3.10.3</span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text">Hardy & Wright, § 22.13</span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text">Hardy & Wright, Thm. 329</span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text">Hardy & Wright, Thms. 271, 272</span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text">Hardy & Wright, eq. 16.3.1</span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text">Ramanujan, <i>Some Formulæ in the Analytic Theory of Numbers</i>, eq. (C); <i>Papers</i> p. 133. A footnote says that Hardy told Ramanujan it also appears in an 1857 paper by Liouville.</span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text">Ramanujan, <i>Some Formulæ in the Analytic Theory of Numbers</i>, eq. (F); <i>Papers</i> p. 134</span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text">Apostol, <i>Modular Functions ...</i>, ch. 6 eq. 4</span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text">Apostol, <i>Modular Functions ...</i>, ch. 6 eq. 3</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=41" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTom_M._Apostol1976" class="citation cs2"><a href="/wiki/Tom_M._Apostol" title="Tom M. Apostol">Tom M. Apostol</a> (1976), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoan00apos_0"><i>Introduction to Analytic Number Theory</i></a></span>, Springer <a href="/wiki/Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90163-9" title="Special:BookSources/0-387-90163-9"><bdi>0-387-90163-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Analytic+Number+Theory&rft.pub=Springer+Undergraduate+Texts+in+Mathematics&rft.date=1976&rft.isbn=0-387-90163-9&rft.au=Tom+M.+Apostol&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoan00apos_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApostol1989" class="citation cs2"><a href="/wiki/Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (1989), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/modularfunctions0000apos"><i>Modular Functions and Dirichlet Series in Number Theory (2nd Edition)</i></a></span>, New York: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-97127-0" title="Special:BookSources/0-387-97127-0"><bdi>0-387-97127-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modular+Functions+and+Dirichlet+Series+in+Number+Theory+%282nd+Edition%29&rft.place=New+York&rft.pub=Springer&rft.date=1989&rft.isbn=0-387-97127-0&rft.aulast=Apostol&rft.aufirst=Tom+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmodularfunctions0000apos&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBatemanDiamond2004" class="citation cs2"><a href="/wiki/Paul_T._Bateman" title="Paul T. Bateman">Bateman, Paul T.</a>; Diamond, Harold G. (2004), <i>Analytic number theory, an introduction</i>, <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-238-938-1" title="Special:BookSources/978-981-238-938-1"><bdi>978-981-238-938-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytic+number+theory%2C+an+introduction&rft.pub=World+Scientific&rft.date=2004&rft.isbn=978-981-238-938-1&rft.aulast=Bateman&rft.aufirst=Paul+T.&rft.au=Diamond%2C+Harold+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen1993" class="citation cs2"><a href="/wiki/Henri_Cohen_(number_theorist)" title="Henri Cohen (number theorist)">Cohen, Henri</a> (1993), <i>A Course in Computational Algebraic Number Theory</i>, Berlin: <a href="/wiki/Springer_Nature" title="Springer Nature">Springer</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-55640-0" title="Special:BookSources/3-540-55640-0"><bdi>3-540-55640-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Computational+Algebraic+Number+Theory&rft.place=Berlin&rft.pub=Springer&rft.date=1993&rft.isbn=3-540-55640-0&rft.aulast=Cohen&rft.aufirst=Henri&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdwards1977" class="citation book cs1"><a href="/wiki/Harold_Edwards_(mathematician)" title="Harold Edwards (mathematician)">Edwards, Harold</a> (1977). <i>Fermat's Last Theorem</i>. New York: <a href="/wiki/Springer_Nature" title="Springer Nature">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90230-9" title="Special:BookSources/0-387-90230-9"><bdi>0-387-90230-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fermat%27s+Last+Theorem&rft.place=New+York&rft.pub=Springer&rft.date=1977&rft.isbn=0-387-90230-9&rft.aulast=Edwards&rft.aufirst=Harold&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardy1999" class="citation cs2"><a href="/wiki/G._H._Hardy" title="G. H. Hardy">Hardy, G. H.</a> (1999), <i>Ramanujan: Twelve Lectures on Subjects Suggested by his Life and work</i>, Providence RI: AMS / Chelsea, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10115%2F1436">10115/1436</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-2023-0" title="Special:BookSources/978-0-8218-2023-0"><bdi>978-0-8218-2023-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%3A+Twelve+Lectures+on+Subjects+Suggested+by+his+Life+and+work&rft.place=Providence+RI&rft.pub=AMS+%2F+Chelsea&rft.date=1999&rft_id=info%3Ahdl%2F10115%2F1436&rft.isbn=978-0-8218-2023-0&rft.aulast=Hardy&rft.aufirst=G.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardyWright1979" class="citation book cs1"><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) [1938]. <i>An Introduction to the Theory of Numbers</i> (5th ed.). Oxford: Clarendon Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-853171-0" title="Special:BookSources/0-19-853171-0"><bdi>0-19-853171-0</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0568909">0568909</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:0423.10001">0423.10001</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=5th&rft.pub=Clarendon+Press&rft.date=1979&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0423.10001%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0568909%23id-name%3DMR&rft.isbn=0-19-853171-0&rft.aulast=Hardy&rft.aufirst=G.+H.&rft.au=Wright%2C+E.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJameson2003" class="citation cs2">Jameson, G. J. O. (2003), <i>The Prime Number Theorem</i>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-89110-8" title="Special:BookSources/0-521-89110-8"><bdi>0-521-89110-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Prime+Number+Theorem&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=0-521-89110-8&rft.aulast=Jameson&rft.aufirst=G.+J.+O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoblitz1984" class="citation cs2">Koblitz, Neal (1984), <i>Introduction to Elliptic Curves and Modular Forms</i>, New York: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-97966-2" title="Special:BookSources/0-387-97966-2"><bdi>0-387-97966-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Elliptic+Curves+and+Modular+Forms&rft.place=New+York&rft.pub=Springer&rft.date=1984&rft.isbn=0-387-97966-2&rft.aulast=Koblitz&rft.aufirst=Neal&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandau1966" class="citation cs2"><a href="/wiki/Edmund_Landau" title="Edmund Landau">Landau, Edmund</a> (1966), <i>Elementary Number Theory</i>, New York: Chelsea</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Number+Theory&rft.place=New+York&rft.pub=Chelsea&rft.date=1966&rft.aulast=Landau&rft.aufirst=Edmund&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliam_J._LeVeque1996" class="citation cs2"><a href="/wiki/William_J._LeVeque" title="William J. LeVeque">William J. LeVeque</a> (1996), <i>Fundamentals of Number Theory</i>, Courier Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-68906-9" title="Special:BookSources/0-486-68906-9"><bdi>0-486-68906-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Number+Theory&rft.pub=Courier+Dover+Publications&rft.date=1996&rft.isbn=0-486-68906-9&rft.au=William+J.+LeVeque&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLong1972" class="citation cs2">Long, Calvin T. (1972), <i>Elementary Introduction to Number Theory</i> (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%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFElliott_Mendelson1987" class="citation cs2">Elliott Mendelson (1987), <i>Introduction to Mathematical Logic</i>, CRC Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-412-80830-7" title="Special:BookSources/0-412-80830-7"><bdi>0-412-80830-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Mathematical+Logic&rft.pub=CRC+Press&rft.date=1987&rft.isbn=0-412-80830-7&rft.au=Elliott+Mendelson&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNagell1964" class="citation cs2"><a href="/wiki/Trygve_Nagell" title="Trygve Nagell">Nagell, Trygve</a> (1964), <i>Introduction to number theory (2nd Edition)</i>, Chelsea, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-2833-5" title="Special:BookSources/978-0-8218-2833-5"><bdi>978-0-8218-2833-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+number+theory+%282nd+Edition%29&rft.pub=Chelsea&rft.date=1964&rft.isbn=978-0-8218-2833-5&rft.aulast=Nagell&rft.aufirst=Trygve&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNivenZuckerman1972" class="citation cs2"><a href="/wiki/Ivan_M._Niven" title="Ivan M. Niven">Niven, Ivan M.</a>; Zuckerman, Herbert S. (1972), <i>An introduction to the theory of numbers (3rd Edition)</i>, <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-64154-5" title="Special:BookSources/0-471-64154-5"><bdi>0-471-64154-5</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+%283rd+Edition%29&rft.pub=John+Wiley+%26+Sons&rft.date=1972&rft.isbn=0-471-64154-5&rft.aulast=Niven&rft.aufirst=Ivan+M.&rft.au=Zuckerman%2C+Herbert+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></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), <i>Elements of Number Theory</i>, 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%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan2000" class="citation cs2">Ramanujan, Srinivasa (2000), <i>Collected Papers</i>, Providence RI: AMS / Chelsea, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-2076-6" title="Special:BookSources/978-0-8218-2076-6"><bdi>978-0-8218-2076-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Collected+Papers&rft.place=Providence+RI&rft.pub=AMS+%2F+Chelsea&rft.date=2000&rft.isbn=978-0-8218-2076-6&rft.aulast=Ramanujan&rft.aufirst=Srinivasa&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliams2011" class="citation cs2">Williams, Kenneth S. (2011), <i>Number theory in the spirit of Liouville</i>, London Mathematical Society Student Texts, vol. 76, Cambridge: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-17562-3" title="Special:BookSources/978-0-521-17562-3"><bdi>978-0-521-17562-3</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:1227.11002">1227.11002</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+theory+in+the+spirit+of+Liouville&rft.place=Cambridge&rft.series=London+Mathematical+Society+Student+Texts&rft.pub=Cambridge+University+Press&rft.date=2011&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1227.11002%23id-name%3DZbl&rft.isbn=978-0-521-17562-3&rft.aulast=Williams&rft.aufirst=Kenneth+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=42" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwarzSpilker1994" class="citation cs2">Schwarz, Wolfgang; Spilker, Jürgen (1994), <i>Arithmetical Functions. An introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic properties</i>, London Mathematical Society Lecture Note Series, vol. 184, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-42725-8" title="Special:BookSources/0-521-42725-8"><bdi>0-521-42725-8</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:0807.11001">0807.11001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Arithmetical+Functions.+An+introduction+to+elementary+and+analytic+properties+of+arithmetic+functions+and+to+some+of+their+almost-periodic+properties&rft.series=London+Mathematical+Society+Lecture+Note+Series&rft.pub=Cambridge+University+Press&rft.date=1994&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0807.11001%23id-name%3DZbl&rft.isbn=0-521-42725-8&rft.aulast=Schwarz&rft.aufirst=Wolfgang&rft.au=Spilker%2C+J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic_function&action=edit&section=43" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Arithmetic_function">"Arithmetic function"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <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=Arithmetic+function&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DArithmetic_function&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic+function" class="Z3988"></span></li> <li>Matthew Holden, Michael Orrison, Michael Varble <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160305132124/https://www.math.hmc.edu/~orrison/research/papers/totient.pdf">Yet another Generalization of Euler's Totient Function</a></li> <li>Huard, Ou, Spearman, and Williams. <a rel="nofollow" class="external text" href="http://mathstat.carleton.ca/~williams/papers/pdf/249.pdf">Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions</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>László Tóth, <a rel="nofollow" class="external text" href="https://arxiv.org/PS_cache/arxiv/pdf/1103/1103.5861v2.pdf">Menon's Identity and arithmetical sums representing functions of several variables</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · 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numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Powers_and_related_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Exponentiation" title="Exponentiation">Powers</a> and related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Power of 2</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Power of 3</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Power of 10</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cube</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Fourth power</a></li> <li><a href="/wiki/Fifth_power_(algebra)" title="Fifth power (algebra)">Fifth power</a></li> <li><a href="/wiki/Sixth_power" title="Sixth power">Sixth power</a></li> <li><a href="/wiki/Seventh_power" title="Seventh power">Seventh power</a></li> <li><a href="/wiki/Eighth_power" title="Eighth power">Eighth power</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Prime_power" title="Prime power">Prime power</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1" style="font-size:114%;margin:0 4em">Of the form <i>a</i> × 2<sup><i>b</i></sup> ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cullen_number" title="Cullen number">Cullen</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li> <li><a href="/wiki/Fermat_number" title="Fermat number">Fermat</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne</a></li> <li><a href="/wiki/Proth_number" class="mw-redirect" title="Proth number">Proth</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hilbert_number" title="Hilbert number">Hilbert</a></li> <li><a href="/wiki/Idoneal_number" title="Idoneal number">Idoneal</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland</a></li> <li><a href="/wiki/Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li> <li><a href="/wiki/Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li> <li><a href="/wiki/Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li> <li><a href="/wiki/Leonardo_number" title="Leonardo number">Leonardo</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas</a></li> <li><a href="/wiki/Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li> <li><a href="/wiki/Padovan_sequence" title="Padovan sequence">Padovan</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell</a></li> <li><a href="/wiki/Perrin_number" title="Perrin number">Perrin</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amenable_number" title="Amenable number">Amenable</a></li> <li><a href="/wiki/Congruent_number" title="Congruent number">Congruent</a></li> <li><a href="/wiki/Kn%C3%B6del_number" title="Knödel number">Knödel</a></li> <li><a href="/wiki/Riesel_number" title="Riesel number">Riesel</a></li> <li><a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li> <li><a href="/wiki/Polite_number" title="Polite number">Polite</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li> <li><a href="/wiki/Ulam_number" title="Ulam number">Ulam</a></li> <li><a href="/wiki/Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li> <li><a href="/wiki/Centered_square_number" title="Centered square number">Centered square</a></li> <li><a href="/wiki/Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li> <li><a href="/wiki/Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li> <li><a href="/wiki/Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li> <li><a href="/wiki/Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li> <li><a href="/wiki/Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li> <li><a href="/wiki/Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li> <li><a href="/wiki/Star_number" title="Star number">Star</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triangular_number" title="Triangular number">Triangular</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Square_triangular_number" title="Square triangular number">Square triangular</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal</a></li> <li><a href="/wiki/Octagonal_number" title="Octagonal number">Octagonal</a></li> <li><a href="/wiki/Nonagonal_number" title="Nonagonal number">Nonagonal</a></li> <li><a href="/wiki/Decagonal_number" title="Decagonal number">Decagonal</a></li> <li><a href="/wiki/Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li> <li><a href="/wiki/Centered_cube_number" title="Centered cube number">Centered cube</a></li> <li><a href="/wiki/Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li> <li><a href="/wiki/Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li> <li><a href="/wiki/Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic</a></li> <li><a href="/wiki/Octahedral_number" title="Octahedral number">Octahedral</a></li> <li><a href="/wiki/Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li> <li><a href="/wiki/Icosahedral_number" title="Icosahedral number">Icosahedral</a></li> <li><a href="/wiki/Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentatope_number" title="Pentatope number">Pentatope</a></li> <li><a href="/wiki/Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Tesseractic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_number" title="Bell number">Bell</a></li> <li><a href="/wiki/Cake_number" title="Cake number">Cake</a></li> <li><a href="/wiki/Catalan_number" title="Catalan number">Catalan</a></li> <li><a href="/wiki/Dedekind_number" title="Dedekind number">Dedekind</a></li> <li><a href="/wiki/Delannoy_number" title="Delannoy number">Delannoy</a></li> <li><a href="/wiki/Euler_number" class="mw-redirect" title="Euler number">Euler</a></li> <li><a href="/wiki/Eulerian_number" title="Eulerian number">Eulerian</a></li> <li><a href="/wiki/Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li> <li><a href="/wiki/Lah_number" title="Lah number">Lah</a></li> <li><a href="/wiki/Lazy_caterer%27s_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li> <li><a href="/wiki/Lobb_number" title="Lobb number">Lobb</a></li> <li><a href="/wiki/Motzkin_number" title="Motzkin number">Motzkin</a></li> <li><a href="/wiki/Narayana_number" title="Narayana number">Narayana</a></li> <li><a href="/wiki/Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li> <li><a href="/wiki/Schr%C3%B6der_number" title="Schröder number">Schröder</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li> <li><a href="/wiki/Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling first</a></li> <li><a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li> <li><a href="/wiki/Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li> <li><a href="/wiki/Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes" style="font-size:114%;margin:0 4em"><a href="/wiki/Prime_number" title="Prime number">Primes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</a></li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a></li> <li><a href="/wiki/Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes" style="font-size:114%;margin:0 4em"><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li> <li><a href="/wiki/Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Arithmetic functions</a> and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Divisor_function" title="Divisor function">Divisor functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_omega_function" title="Prime omega function">Prime omega functions</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Highly_totient_number" title="Highly totient number">Highly totient</a></li> <li><a href="/wiki/Noncototient" title="Noncototient">Noncototient</a></li> <li><a href="/wiki/Nontotient" title="Nontotient">Nontotient</a></li> <li><a href="/wiki/Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li> <li><a href="/wiki/Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amicable_numbers" title="Amicable numbers">Amicable</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Sociable_numbers" class="mw-redirect" title="Sociable numbers">Sociable</a></li> <li><a href="/wiki/Untouchable_number" title="Untouchable number">Untouchable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Primorial" title="Primorial">Primorial</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euclid_number" title="Euclid number">Euclid</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_prime_factor_or_divisor_related_numbers" style="font-size:114%;margin:0 4em">Other <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> or <a href="/wiki/Divisor" title="Divisor">divisor</a> related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blum_integer" title="Blum integer">Blum</a></li> <li><a href="/wiki/Cyclic_number_(group_theory)" title="Cyclic number (group theory)">Cyclic</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Woods_number" title="Erdős–Woods number">Erdős–Woods</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Giuga_number" title="Giuga number">Giuga</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Jordan%E2%80%93P%C3%B3lya_number" title="Jordan–Pólya number">Jordan–Pólya</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a href="/wiki/Regular_number" title="Regular number">Regular</a></li> <li><a href="/wiki/Rough_number" title="Rough number">Rough</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/St%C3%B8rmer_number" title="Størmer number">Størmer</a></li> <li><a href="/wiki/Super-Poulet_number" title="Super-Poulet number">Super-Poulet</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numeral_system-dependent_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Numeral_system" title="Numeral system">Numeral system</a>-dependent numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Arithmetic functions</a> <br />and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Persistence_of_a_number" title="Persistence of a number">Persistence</a> <ul><li><a href="/wiki/Additive_persistence" class="mw-redirect" title="Additive persistence">Additive</a></li> <li><a href="/wiki/Multiplicative_persistence" class="mw-redirect" title="Multiplicative persistence">Multiplicative</a></li></ul></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</a></li> <li><a href="/wiki/Digital_root" title="Digital root">Digital root</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit product</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Multiplicative_digital_root" title="Multiplicative digital root">Multiplicative digital root</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coding-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Meertens_number" title="Meertens number">Meertens</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dudeney_number" title="Dudeney number">Dudeney</a></li> <li><a href="/wiki/Factorion" title="Factorion">Factorion</a></li> <li><a href="/wiki/Kaprekar_number" title="Kaprekar number">Kaprekar</a></li> <li><a href="/wiki/Kaprekar%27s_routine" title="Kaprekar's routine">Kaprekar's constant</a></li> <li><a href="/wiki/Keith_number" title="Keith number">Keith</a></li> <li><a href="/wiki/Lychrel_number" title="Lychrel number">Lychrel</a></li> <li><a href="/wiki/Narcissistic_number" title="Narcissistic number">Narcissistic</a></li> <li><a href="/wiki/Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li> <li><a href="/wiki/Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a> <ul><li><a href="/wiki/Happy_number" title="Happy number">Happy</a></li></ul></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Automorphic_number" title="Automorphic number">Automorphic</a> <ul><li><a href="/wiki/Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_digit" title="Numerical digit">Digit</a>-composition related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Palindromic_number" title="Palindromic number">Palindromic</a></li> <li><a href="/wiki/Pandigital_number" title="Pandigital number">Pandigital</a></li> <li><a href="/wiki/Repdigit" title="Repdigit">Repdigit</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit</a></li> <li><a href="/wiki/Self-descriptive_number" title="Self-descriptive number">Self-descriptive</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_number" title="Smarandache–Wellin number">Smarandache–Wellin</a></li> <li><a href="/wiki/Undulating_number" title="Undulating number">Undulating</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit-<a href="/wiki/Permutation" title="Permutation">permutation</a> related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cyclic_number" title="Cyclic number">Cyclic</a></li> <li><a href="/wiki/Digit-reassembly_number" title="Digit-reassembly number">Digit-reassembly</a></li> <li><a href="/wiki/Parasitic_number" title="Parasitic number">Parasitic</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Transposable_integer" title="Transposable integer">Transposable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisor-related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li> <li><a href="/wiki/Vampire_number" title="Vampire number">Vampire</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Friedman_number" title="Friedman number">Friedman</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Binary_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Binary_number" title="Binary number">Binary numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Evil_number" title="Evil number">Evil</a></li> <li><a href="/wiki/Odious_number" title="Odious number">Odious</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve" style="font-size:114%;margin:0 4em">Generated via a <a href="/wiki/Sieve_theory" title="Sieve theory">sieve</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Generation_of_primes" title="Generation of primes">Prime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pancake_sorting" title="Pancake sorting">Pancake number</a></li> <li><a href="/wiki/Sorting_number" title="Sorting number">Sorting number</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Natural_language" title="Natural language">Natural language</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aronson%27s_sequence" title="Aronson's sequence">Aronson's sequence</a></li> <li><a href="/wiki/Ban_number" title="Ban number">Ban</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Strobogrammatic_number" title="Strobogrammatic number">Strobogrammatic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td 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Arithmetic function - Wikipedia