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<title>Möbius function</title>

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Möbius function

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the number-theoretic Möbius function. For the combinatorial Möbius function, see <a href="Incidence_algebra" title="Incidence algebra">incidence algebra</a>. For the <a href="Rational_function" title="Rational function">rational functions</a> defined on the <a href="Complex_number" title="Complex number">complex numbers</a>, see <a href="M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformation</a>.</div>

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</style><table class="infobox vcard"><caption class="infobox-title fn org">Möbius function</caption><tbody><tr><th scope="row" class="infobox-label">Named after</th><td class="infobox-data"><a href="August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">August Ferdinand Möbius</a></td></tr><tr><th scope="row" class="infobox-label">Publication year</th><td class="infobox-data">1832</td></tr><tr><th scope="row" class="infobox-label">Author of publication</th><td class="infobox-data"><a href="August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">August Ferdinand Möbius</a></td></tr><tr><th scope="row" class="infobox-label"><abbr title="Number">No.</abbr> of known terms</th><td class="infobox-data">infinite</td></tr><tr><th scope="row" class="infobox-label">First terms</th><td class="infobox-data">1, −1, −1, 0, −1, 1, −1, 0, 0, 1</td></tr><tr><th scope="row" class="infobox-label"><a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a> index</th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">

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</style><div class="plainlist"><ul><li><a rel="nofollow" class="external text" href="https://oeis.org/A008683">A008683</a></li><li>Möbius (or Moebius) function mu(n). mu(1) = 1; mu(n) = (-1)^k if n is the product of k different primes; otherwise mu(n) = 0.</li></ul></div></td></tr></tbody></table>

The Möbius function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \mu (n)}</annotation>

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</math><img src="./6412d2052c8e54a65d596a419b429d00fd18d53b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.606ex; height:2.843ex;" alt="{\displaystyle \mu (n)}" loading="lazy"> is a <a href="Multiplicative_function" title="Multiplicative function">multiplicative function</a> in <a href="Number_theory" title="Number theory">number theory</a> introduced by the German mathematician <a href="August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">August Ferdinand Möbius</a> (also transliterated Moebius) in 1832.<a href="#cite_note-1">[i]</a><a href="#cite_note-3">[ii]</a><a href="#cite_note-FOOTNOTEMöbius1832105–123-4">[2]</a> It is ubiquitous in elementary and <a href="Analytic_number_theory" title="Analytic number theory">analytic number theory</a> and most often appears as part of its namesake the <a href="M%C3%B6bius_inversion_formula" title="Möbius inversion formula">Möbius inversion formula</a>. Following work of <a href="Gian-Carlo_Rota" title="Gian-Carlo Rota">Gian-Carlo Rota</a> in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \mu (x)}</annotation>

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</math><img src="./f339251a09ebf15dd50bb751d27b02820f68c545.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle \mu (x)}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

The Möbius function is defined by<a href="#cite_note-FOOTNOTEAbramowitzStegun1972826-5">[3]</a>

<dl><dd><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (n)={\begin{cases}1&{\text{if }}n=1\\(-1)^{k}&{\text{if }}n{\text{ is the product of }}k{\text{ distinct primes}}\\0&{\text{if }}n{\text{ is divisible by a square}}>1\end{cases}}}">

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

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<mtr>

<mtd>

<msup>

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</msup>

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<mtd>

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<mtext> is the product of </mtext>

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<mtext> distinct primes</mtext>

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</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

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<mtext> is divisible by a square</mtext>

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<annotation encoding="application/x-tex">{\displaystyle \mu (n)={\begin{cases}1&{\text{if }}n=1\\(-1)^{k}&{\text{if }}n{\text{ is the product of }}k{\text{ distinct primes}}\\0&{\text{if }}n{\text{ is divisible by a square}}>1\end{cases}}}</annotation>

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The Möbius function can alternatively be represented as

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<annotation encoding="application/x-tex">{\displaystyle \mu (n)=\delta _{\omega (n)\Omega (n)}\lambda (n),}</annotation>

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</math><img src="./e688a6d05a27bcf0942463c2c78588d6a9458c86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.915ex; height:3.176ex;" alt="{\displaystyle \mu (n)=\delta _{\omega (n)\Omega (n)}\lambda (n),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation>

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</math><img src="./fa75d04c11480d976e1396951e02cbb3c4f71568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}" loading="lazy"> is the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda (n)}</annotation>

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</math><img src="./8f06b26d7cb84392c1891a14c32c4dbe7e3f5e92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.559ex; height:2.843ex;" alt="{\displaystyle \lambda (n)}" loading="lazy"> is the <a href="Liouville_function" title="Liouville function">Liouville function</a>,

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \omega (n)}</annotation>

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</math><img src="./f9e1e190faf1ba7931e4e98594df6097d297bed1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.65ex; height:2.843ex;" alt="{\displaystyle \omega (n)}" loading="lazy"></a> is the number of distinct prime divisors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, and <a href="Prime_omega_function" title="Prime omega function"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \Omega (n)}</annotation>

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</math><img src="./6becc31c61ad3420a1e4ee9e39c28baf73bda24d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle \Omega (n)}" loading="lazy"></a> is the number of prime factors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, counted with multiplicity.

Another characterization by <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> is the sum of all <a href="Primitive_root_modulo_n" title="Primitive root modulo n">primitive roots</a>.<a href="#cite_note-6">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Values">Values</h2></div>

The values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (n)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mu (n)}</annotation>

</semantics>

</math><img src="./6412d2052c8e54a65d596a419b429d00fd18d53b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.606ex; height:2.843ex;" alt="{\displaystyle \mu (n)}" loading="lazy"> for the first 50 positive numbers are

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">

</th>

<th style="width: 30px;">1

</th>

<th style="width: 30px;">2

</th>

<th style="width: 30px;">3

</th>

<th style="width: 30px;">4

</th>

<th style="width: 30px;">5

</th>

<th style="width: 30px;">6

</th>

<th style="width: 30px;">7

</th>

<th style="width: 30px;">8

</th>

<th style="width: 30px;">9

</th>

<th style="width: 30px;">10

</th></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)}</annotation>

</semantics>

</math><img src="./6412d2052c8e54a65d596a419b429d00fd18d53b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.606ex; height:2.843ex;" alt="{\displaystyle \mu (n)}" loading="lazy">

</th>

<td>1

</td>

<td>−1

</td>

<td>−1

</td>

<td>0

</td>

<td>−1

</td>

<td>1

</td>

<td>−1

</td>

<td>0

</td>

<td>0

</td>

<td>1

</td></tr></tbody></table>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">

</th>

<th style="width: 30px;">11

</th>

<th style="width: 30px;">12

</th>

<th style="width: 30px;">13

</th>

<th style="width: 30px;">14

</th>

<th style="width: 30px;">15

</th>

<th style="width: 30px;">16

</th>

<th style="width: 30px;">17

</th>

<th style="width: 30px;">18

</th>

<th style="width: 30px;">19

</th>

<th style="width: 30px;">20

</th></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)}</annotation>

</semantics>

</math><img src="./6412d2052c8e54a65d596a419b429d00fd18d53b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.606ex; height:2.843ex;" alt="{\displaystyle \mu (n)}" loading="lazy">

</th>

<td>−1

</td>

<td>0

</td>

<td>−1

</td>

<td>1

</td>

<td>1

</td>

<td>0

</td>

<td>−1

</td>

<td>0

</td>

<td>−1

</td>

<td>0

</td></tr></tbody></table>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">

</th>

<th style="width: 30px;">21

</th>

<th style="width: 30px;">22

</th>

<th style="width: 30px;">23

</th>

<th style="width: 30px;">24

</th>

<th style="width: 30px;">25

</th>

<th style="width: 30px;">26

</th>

<th style="width: 30px;">27

</th>

<th style="width: 30px;">28

</th>

<th style="width: 30px;">29

</th>

<th style="width: 30px;">30

</th></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)}</annotation>

</semantics>

</math><img src="./6412d2052c8e54a65d596a419b429d00fd18d53b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.606ex; height:2.843ex;" alt="{\displaystyle \mu (n)}" loading="lazy">

</th>

<td>1

</td>

<td>1

</td>

<td>−1

</td>

<td>0

</td>

<td>0

</td>

<td>1

</td>

<td>0

</td>

<td>0

</td>

<td>−1

</td>

<td>−1

</td></tr></tbody></table>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">

</th>

<th style="width: 30px;">31

</th>

<th style="width: 30px;">32

</th>

<th style="width: 30px;">33

</th>

<th style="width: 30px;">34

</th>

<th style="width: 30px;">35

</th>

<th style="width: 30px;">36

</th>

<th style="width: 30px;">37

</th>

<th style="width: 30px;">38

</th>

<th style="width: 30px;">39

</th>

<th style="width: 30px;">40

</th></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)}</annotation>

</semantics>

</math><img src="./6412d2052c8e54a65d596a419b429d00fd18d53b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.606ex; height:2.843ex;" alt="{\displaystyle \mu (n)}" loading="lazy">

</th>

<td>−1

</td>

<td>0

</td>

<td>1

</td>

<td>1

</td>

<td>1

</td>

<td>0

</td>

<td>−1

</td>

<td>1

</td>

<td>1

</td>

<td>0

</td></tr></tbody></table>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">

</th>

<th style="width: 30px;">41

</th>

<th style="width: 30px;">42

</th>

<th style="width: 30px;">43

</th>

<th style="width: 30px;">44

</th>

<th style="width: 30px;">45

</th>

<th style="width: 30px;">46

</th>

<th style="width: 30px;">47

</th>

<th style="width: 30px;">48

</th>

<th style="width: 30px;">49

</th>

<th style="width: 30px;">50

</th></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)}</annotation>

</semantics>

</math><img src="./6412d2052c8e54a65d596a419b429d00fd18d53b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.606ex; height:2.843ex;" alt="{\displaystyle \mu (n)}" loading="lazy">

</th>

<td>−1

</td>

<td>−1

</td>

<td>−1

</td>

<td>0

</td>

<td>0

</td>

<td>1

</td>

<td>−1

</td>

<td>0

</td>

<td>0

</td>

<td>0

</td></tr></tbody></table>

The first 50 values of the function are plotted below:

Larger values can be checked in:

<ul><li><a rel="nofollow" class="external text" href="https://www.wolframalpha.com/input/?i=MoebiusMu+123">Wolframalpha</a></li>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Mathematical_series">Mathematical series</h3></div>

The <a href="Dirichlet_series" title="Dirichlet series">Dirichlet series</a> that <a href="Generating_function" title="Generating function">generates</a> the Möbius function is the (multiplicative) inverse of the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>; if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> is a complex number with real part larger than 1 we have

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}={\frac {1}{\zeta (s)}}.}</annotation>

</semantics>

</math><img src="./8173bcb43b3e46e7ed72d57ac9d6ad1c2b7391fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.76ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}={\frac {1}{\zeta (s)}}.}" loading="lazy"></dd></dl>

This may be seen from its <a href="Euler_product" title="Euler product">Euler product</a>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mtext> prime</mtext>

</mrow>

</munder>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\zeta (s)}}=\prod _{p{\text{ prime}}}{\left(1-{\frac {1}{p^{s}}}\right)}=\left(1-{\frac {1}{2^{s}}}\right)\left(1-{\frac {1}{3^{s}}}\right)\left(1-{\frac {1}{5^{s}}}\right)\cdots }</annotation>

</semantics>

</math><img src="./671eb68df48a3f2dd9358c39d1cb81e109d623d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:62.533ex; height:7.009ex;" alt="{\displaystyle {\frac {1}{\zeta (s)}}=\prod _{p{\text{ prime}}}{\left(1-{\frac {1}{p^{s}}}\right)}=\left(1-{\frac {1}{2^{s}}}\right)\left(1-{\frac {1}{3^{s}}}\right)\left(1-{\frac {1}{5^{s}}}\right)\cdots }" loading="lazy"></dd></dl>

Also:

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum \limits _{n=1}^{\infty }{\frac {|\mu (n)|}{n^{s}}}={\frac {\zeta (s)}{\zeta (2s)}};}</annotation>

</semantics>

</math><img src="./05fcf46da42b3cacc9f6ee2a5e78b35994d081cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.216ex; height:6.843ex;" alt="{\displaystyle \sum \limits _{n=1}^{\infty }{\frac {|\mu (n)|}{n^{s}}}={\frac {\zeta (s)}{\zeta (2s)}};}" loading="lazy"></li>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}=0;}</annotation>

</semantics>

</math><img src="./afad097b27b39ba27c71d164d745436efe8dab3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.092ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}=0;}" loading="lazy"></li>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum \limits _{n=1}^{\infty }{\frac {\mu (n)\ln n}{n}}=-1;}</annotation>

</semantics>

</math><img src="./f39d4cd87f963cce90dc191c3a9ed80bfc2ecd99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.008ex; height:6.843ex;" alt="{\displaystyle \sum \limits _{n=1}^{\infty }{\frac {\mu (n)\ln n}{n}}=-1;}" loading="lazy"></li>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum \limits _{n=1}^{\infty }{\frac {\mu (n)\ln ^{2}n}{n}}=-2\gamma ,}</annotation>

</semantics>

</math><img src="./d3952f4d4cf42adadd18f03300612d88e3dcb452.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.325ex; height:7.176ex;" alt="{\displaystyle \sum \limits _{n=1}^{\infty }{\frac {\mu (n)\ln ^{2}n}{n}}=-2\gamma ,}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> is <a href="Euler's_constant" title="Euler's constant">Euler's constant</a>.</li></ul>

The <a href="Lambert_series" title="Lambert series">Lambert series</a> for the Möbius function is

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)q^{n}}{1-q^{n}}}=q,}</annotation>

</semantics>

</math><img src="./1d9234c2f32243af3dc02aef2edd9d4cb0e9fbfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.297ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)q^{n}}{1-q^{n}}}=q,}" loading="lazy"></dd></dl>

which converges for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |q|<1}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |q|<1}</annotation>

</semantics>

</math><img src="./ef46eeba8cb2497a797ec6652c0f76d69e2175bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.624ex; height:2.843ex;" alt="{\displaystyle |q|<1}" loading="lazy">. For prime <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \geq 2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \geq 2}</annotation>

</semantics>

</math><img src="./be8fc8f0b74bf5629d672d0d1d73719fc1618c6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.749ex; height:2.343ex;" alt="{\displaystyle \alpha \geq 2}" loading="lazy">, we also have

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (\alpha n)q^{n}}{q^{n}-1}}=\sum _{n\geq 0}q^{\alpha ^{n}},|q|<1.}</annotation>

</semantics>

</math><img src="./897807547f7286238a8f8a56b2ed4203bde9c642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.443ex; height:7.009ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (\alpha n)q^{n}}{q^{n}-1}}=\sum _{n\geq 0}q^{\alpha ^{n}},|q|<1.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Algebraic_number_theory">Algebraic number theory</h3></div>

Gauss<a href="#cite_note-FOOTNOTEGauss1986Art._81-2">[1]</a> proved that for a prime number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> the sum of its <a href="Primitive_root_modulo_n#Arithmetic_facts" title="Primitive root modulo n">primitive roots</a> is congruent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (p-1)\mod p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (p-1)\mod p}</annotation>

</semantics>

</math><img src="./4862f70a844e16be1223596c6fe16c4274566ec0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.815ex; height:2.843ex;" alt="{\displaystyle \mu (p-1)\mod p}" loading="lazy">.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}}</annotation>

</semantics>

</math><img src="./dbb96e056c071d13fc7702013f9273e7f5cd88a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}" loading="lazy"> denotes the <a href="Finite_field" title="Finite field">finite field</a> of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> is necessarily a <a href="Prime_power" title="Prime power">prime power</a>), then the number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> of monic irreducible polynomials of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}}</annotation>

</semantics>

</math><img src="./dbb96e056c071d13fc7702013f9273e7f5cd88a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}" loading="lazy"> is given by<a href="#cite_note-FOOTNOTEJacobson2009§4.13-7">[5]</a>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munder>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(q,n)={\frac {1}{n}}\sum _{d\mid n}\mu (d)q^{\frac {n}{d}}.}</annotation>

</semantics>

</math><img src="./391339633fb9c68b1a28aa20dbcaee3d872bdeb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:24.852ex; height:6.843ex;" alt="{\displaystyle N(q,n)={\frac {1}{n}}\sum _{d\mid n}\mu (d)q^{\frac {n}{d}}.}" loading="lazy"></dd></dl>

The Möbius function is used in the <a href="M%C3%B6bius_inversion_formula" title="Möbius inversion formula">Möbius inversion formula</a>.

<div class="mw-heading mw-heading3"><h3 id="Physics">Physics</h3></div>

The Möbius function also arises in the <a href="Primon_gas" title="Primon gas">primon gas</a> or <a href="Free_Riemann_gas" class="mw-redirect" title="Free Riemann gas">free Riemann gas</a> model of <a href="Supersymmetry" title="Supersymmetry">supersymmetry</a>. In this theory, the fundamental particles or "primons" have energies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log(p)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log(p)}</annotation>

</semantics>

</math><img src="./94a43e7856ef1213d2c4be061246df409ed87a54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.95ex; height:2.843ex;" alt="{\displaystyle \log(p)}" loading="lazy">. Under <a href="Second_quantization" title="Second quantization">second quantization</a>, multiparticle excitations are considered; these are given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log(n)}</annotation>

</semantics>

</math><img src="./503f4ab1a3810662c83bccd74b0d966b642a3f1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.176ex; height:2.843ex;" alt="{\displaystyle \log(n)}" loading="lazy"> for any natural number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">. This follows from the fact that the factorization of the natural numbers into primes is unique.

In the free Riemann gas, any natural number can occur, if the <a href="Primon_gas" title="Primon gas">primons</a> are taken as <a href="Boson" title="Boson">bosons</a>. If they are taken as <a href="Fermion" title="Fermion">fermions</a>, then the <a href="Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a> excludes squares. The operator

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-1)^{F}}</annotation>

</semantics>

</math><img src="./4c263714b06b7135a09cbdec4f1211e9627cd7c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.243ex; height:3.176ex;" alt="{\displaystyle (-1)^{F}}" loading="lazy"></a> that distinguishes fermions and bosons is then none other than the Möbius function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)}</annotation>

</semantics>

</math><img src="./6412d2052c8e54a65d596a419b429d00fd18d53b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.606ex; height:2.843ex;" alt="{\displaystyle \mu (n)}" loading="lazy">.

The free Riemann gas has a number of other interesting connections to number theory, including the fact that the <a href="Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition function</a> is the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>. This idea underlies <a href="Alain_Connes" title="Alain Connes">Alain Connes</a>'s attempted proof of the <a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>.<a href="#cite_note-FOOTNOTEBostConnes1995411–457-8">[6]</a>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

The Möbius function is <a href="Multiplicative_function" title="Multiplicative function">multiplicative</a> (i.e.,

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (ab)=\mu (a)\mu (b)}</annotation>

</semantics>

</math><img src="./5235d215da9800bcea3a3008992c94541a8a0a03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.186ex; height:2.843ex;" alt="{\displaystyle \mu (ab)=\mu (a)\mu (b)}" loading="lazy"> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> are <a href="Coprime" class="mw-redirect" title="Coprime">coprime</a>).

<blockquote>Proof: Given two coprime numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\geq n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\geq n}</annotation>

</semantics>

</math><img src="./6b02f25e62da7fe3162ac80446437cdc1c0fd341.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.534ex; height:2.176ex;" alt="{\displaystyle m\geq n}" loading="lazy">, we induct on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mn}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle mn}</annotation>

</semantics>

</math><img src="./348cd26a0b7a0034f57a951e2cf5f637dd47c1ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:1.676ex;" alt="{\displaystyle mn}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mn=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle mn=1}</annotation>

</semantics>

</math><img src="./3051154b485fa550124f7d3422e6af9c6c8f3ade.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.696ex; height:2.176ex;" alt="{\displaystyle mn=1}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (mn)=1=\mu (m)\mu (n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (mn)=1=\mu (m)\mu (n)}</annotation>

</semantics>

</math><img src="./d53aa9db3b71251229e901f958b8d466abb29d07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.863ex; height:2.843ex;" alt="{\displaystyle \mu (mn)=1=\mu (m)\mu (n)}" loading="lazy">. Otherwise, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m>n\geq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m>n\geq 1}</annotation>

</semantics>

</math><img src="./57c220dabf305c97216137b86371cf85e57299d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.794ex; height:2.343ex;" alt="{\displaystyle m>n\geq 1}" loading="lazy">, so

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</munder>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</munder>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

</mrow>

<mtext>induction</mtext>

</mrow>

</mover>

</mrow>

</mrow>

<msup>

</msup>

</mrow>

</mrow>

</munder>

<msup>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</munder>

<msup>

</msup>

</mrow>

</mrow>

</munder>

<msup>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}0&=\sum _{d|mn}\mu (d)\\&=\mu (mn)+\sum _{d|mn;d<mn}\mu (d)\\&{\stackrel {\text{induction}}{=}}\mu (mn)-\mu (m)\mu (n)+\sum _{d|m;d'|n}\mu (d)\mu (d')\\&=\mu (mn)-\mu (m)\mu (n)+\sum _{d|m}\mu (d)\sum _{d'|n}\mu (d')\\&=\mu (mn)-\mu (m)\mu (n)+0\end{aligned}}}</annotation>

</semantics>

</math></dd></dl>

</blockquote>

The sum of the Möbius function over all positive divisors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> (including <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> itself and 1) is zero except when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=1}</annotation>

</semantics>

</math><img src="./d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy">:

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{d\mid n}\mu (d)={\begin{cases}1&{\text{if }}n=1,\\0&{\text{if }}n>1.\end{cases}}}">

</mrow>

</munder>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{d\mid n}\mu (d)={\begin{cases}1&{\text{if }}n=1,\\0&{\text{if }}n>1.\end{cases}}}</annotation>

</semantics>

</math><img src="./42c8967623869cf210c04030901f1f6d4c3a0619.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.489ex; height:7.176ex;" alt="{\displaystyle \sum _{d\mid n}\mu (d)={\begin{cases}1&{\text{if }}n=1,\\0&{\text{if }}n>1.\end{cases}}}" loading="lazy"></dd></dl>

The equality above leads to the important <a href="M%C3%B6bius_inversion_formula" title="Möbius inversion formula">Möbius inversion formula</a> and is the main reason why <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> is of relevance in the theory of multiplicative and arithmetic functions.

Other applications of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)}</annotation>

</semantics>

</math><img src="./6412d2052c8e54a65d596a419b429d00fd18d53b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.606ex; height:2.843ex;" alt="{\displaystyle \mu (n)}" loading="lazy"> in combinatorics are connected with the use of the <a href="P%C3%B3lya_enumeration_theorem" title="Pólya enumeration theorem">Pólya enumeration theorem</a> in combinatorial groups and combinatorial enumerations.

There is a formula<a href="#cite_note-FOOTNOTEHardyWright1980(16.6.4),_p._239-9">[7]</a> for calculating the Möbius function without directly knowing the factorization of its argument:

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</munder>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)=\sum _{\stackrel {1\leq k\leq n}{\gcd(k,\,n)=1}}e^{2\pi i{\frac {k}{n}}},}</annotation>

</semantics>

</math><img src="./ea6e32b666990ca10ee9d270ac8caba57e6e7d89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:22.555ex; height:8.176ex;" alt="{\displaystyle \mu (n)=\sum _{\stackrel {1\leq k\leq n}{\gcd(k,\,n)=1}}e^{2\pi i{\frac {k}{n}}},}" loading="lazy"></dd></dl>

i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)}</annotation>

</semantics>

</math><img src="./6412d2052c8e54a65d596a419b429d00fd18d53b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.606ex; height:2.843ex;" alt="{\displaystyle \mu (n)}" loading="lazy"> is the sum of the primitive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-th <a href="Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a>. (However, the computational complexity of this definition is at least the same as that of the Euler product definition.)

Other identities satisfied by the Möbius function include

</mrow>

</munder>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k\leq n}\left\lfloor {\frac {n}{k}}\right\rfloor \mu (k)=1}</annotation>

</semantics>

</math><img src="./0a9afb23a44927b7f20593ffebf756fa63a28a41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.498ex; height:6.343ex;" alt="{\displaystyle \sum _{k\leq n}\left\lfloor {\frac {n}{k}}\right\rfloor \mu (k)=1}" loading="lazy"></dd></dl>

and

</mrow>

</munder>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{jk\leq n}\sin \left({\frac {\pi jk}{2}}\right)\mu (k)=1}</annotation>

</semantics>

</math><img src="./03cc808cbdec121eaf9b9003ba30148808370831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.019ex; width:23.502ex; height:7.009ex;" alt="{\displaystyle \sum _{jk\leq n}\sin \left({\frac {\pi jk}{2}}\right)\mu (k)=1}" loading="lazy">.</dd></dl>

The first of these is a classical result while the second was published in 2020.<a href="#cite_note-FOOTNOTEApostol1976-10">[8]</a><a href="#cite_note-FOOTNOTEKline2020-11">[9]</a> Similar identities hold for the <a href="Mertens_function" title="Mertens function">Mertens function</a>.

<div class="mw-heading mw-heading3"><h3 id="Proof_of_the_formula_for_the_sum_of_'"`UNIQ--postMath-00000048-QINU`"'_over_divisors">Proof of the formula for the sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> over divisors</h3></div>

The formula

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{d\mid n}\mu (d)={\begin{cases}1&{\text{if }}n=1,\\0&{\text{if }}n>1\end{cases}}}">

</mrow>

</munder>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{d\mid n}\mu (d)={\begin{cases}1&{\text{if }}n=1,\\0&{\text{if }}n>1\end{cases}}}</annotation>

</semantics>

</math><img src="./521b459898765b0d0753a4abd5c178a5b5356a35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.489ex; height:7.176ex;" alt="{\displaystyle \sum _{d\mid n}\mu (d)={\begin{cases}1&{\text{if }}n=1,\\0&{\text{if }}n>1\end{cases}}}" loading="lazy"></dd></dl>

can be written using <a href="Dirichlet_convolution" title="Dirichlet convolution">Dirichlet convolution</a> as:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1*\mu =\varepsilon }</annotation>

</semantics>

</math><img src="./355fe362ab5f4ece14eca0cf863d10d67d4c1049.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.941ex; height:2.676ex;" alt="{\displaystyle 1*\mu =\varepsilon }" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"> is the <a href="Dirichlet_convolution#Properties" title="Dirichlet convolution">identity under the convolution</a>.

One way of proving this formula is by noting that the Dirichlet convolution of two <a href="Multiplicative_functions" class="mw-redirect" title="Multiplicative functions">multiplicative functions</a> is again multiplicative. Thus it suffices to prove the formula for powers of primes. Indeed, for any prime

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k>0}</annotation>

</semantics>

</math><img src="./27b3af208b148139eefc03f0f80fa94c38c5af45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}" loading="lazy">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1*\mu (p^{k})=\sum _{d\mid p^{k}}\mu (d)=\mu (1)+\mu (p)+\sum _{1<m<=k}\mu (p^{m})=1-1+\sum 0=0=\varepsilon (p^{k})}</annotation>

</semantics>

</math><img src="./1efa0d2b811d161f7fb56f9009343ee4886642c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:79.545ex; height:6.343ex;" alt="{\displaystyle 1*\mu (p^{k})=\sum _{d\mid p^{k}}\mu (d)=\mu (1)+\mu (p)+\sum _{1<m<=k}\mu (p^{m})=1-1+\sum 0=0=\varepsilon (p^{k})}" loading="lazy">,</dd></dl>

while for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=1}</annotation>

</semantics>

</math><img src="./d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1*\mu (1)=\sum _{d\mid 1}\mu (d)=\mu (1)=1=\varepsilon (1)}</annotation>

</semantics>

</math><img src="./205c84136687b0f3a1fd4fc8eedacbc8ae29619d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:37.885ex; height:6.009ex;" alt="{\displaystyle 1*\mu (1)=\sum _{d\mid 1}\mu (d)=\mu (1)=1=\varepsilon (1)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Other_proofs">Other proofs</h4></div>

Another way of proving this formula is by using the identity

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</munder>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)=\sum _{\stackrel {1\leq k\leq n}{\gcd(k,\,n)=1}}e^{2\pi i{\frac {k}{n}}},}</annotation>

</semantics>

</math><img src="./ea6e32b666990ca10ee9d270ac8caba57e6e7d89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:22.555ex; height:8.176ex;" alt="{\displaystyle \mu (n)=\sum _{\stackrel {1\leq k\leq n}{\gcd(k,\,n)=1}}e^{2\pi i{\frac {k}{n}}},}" loading="lazy"></dd></dl>

The formula above is then a consequence of the fact that the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">th roots of unity sum to 0, since each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">th root of unity is a primitive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">th root of unity for exactly one divisor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">.

However it is also possible to prove this identity from first principles. First note that it is trivially true when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=1}</annotation>

</semantics>

</math><img src="./d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy">. Suppose then that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n>1}</annotation>

</semantics>

</math><img src="./ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}" loading="lazy">. Then there is a bijection between the factors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (d)\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (d)\neq 0}</annotation>

</semantics>

</math><img src="./00435ffa0fe3fbedf5bc6ee41135468e2a1172f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.688ex; height:2.843ex;" alt="{\displaystyle \mu (d)\neq 0}" loading="lazy"> and the subsets of the set of all prime factors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">. The asserted result follows from the fact that every non-empty <a href="Finite_set" title="Finite set">finite set</a> has an equal number of odd- and even-cardinality subsets.

This last fact can be shown easily by induction on the cardinality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |S|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |S|}</annotation>

</semantics>

</math><img src="./28d901e98a035ff4c0e37fe6dd8e750ece6c1f0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.793ex; height:2.843ex;" alt="{\displaystyle |S|}" loading="lazy"> of a non-empty finite set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">. First, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |S|=1}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |S|=1}</annotation>

</semantics>

</math><img src="./ea903ffcf577b0c0bc88f78a189ea798d0001a6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.054ex; height:2.843ex;" alt="{\displaystyle |S|=1}" loading="lazy">, there is exactly one odd-cardinality subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">, namely <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> itself, and exactly one even-cardinality subset, namely <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \emptyset }</annotation>

</semantics>

</math><img src="./6af50205f42bb2ec3c666b7b847d2c7f96e464c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }" loading="lazy">. Next, if

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |S|>1}</annotation>

</semantics>

</math><img src="./669eabfcbd5e6b0c12dcfd17dbc448e87c639c08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.054ex; height:2.843ex;" alt="{\displaystyle |S|>1}" loading="lazy">, then divide the subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> into two subclasses depending on whether they contain or not some fixed element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">. There is an obvious bijection between these two subclasses, pairing those subsets that have the same complement relative to the subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x\}}</annotation>

</semantics>

</math><img src="./a120eeb8a091b516595765bd08b306f2394e7721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \{x\}}" loading="lazy">. Also, one of these two subclasses consists of all the subsets of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\setminus \{x\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\setminus \{x\}}</annotation>

</semantics>

</math><img src="./4cc9f366eba8c028439b2b7f1ae392b654271ff9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.349ex; height:2.843ex;" alt="{\displaystyle S\setminus \{x\}}" loading="lazy">, and therefore, by the induction hypothesis, has an equal number of odd- and even-cardinality subsets. These subsets in turn correspond bijectively to the even- and odd-cardinality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x\}}</annotation>

</semantics>

</math><img src="./a120eeb8a091b516595765bd08b306f2394e7721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \{x\}}" loading="lazy">-containing subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">. The inductive step follows directly from these two bijections.

A related result is that the binomial coefficients exhibit alternating entries of odd and even power which sum symmetrically.

<div class="mw-heading mw-heading3"><h3 id="Average_order">Average order</h3></div>

The <a href="Average_order_of_an_arithmetic_function" title="Average order of an arithmetic function">mean value (in the sense of average orders)</a> of the Möbius function is zero. This statement is, in fact, equivalent to the <a href="Prime_number_theorem" title="Prime number theorem">prime number theorem</a>.<a href="#cite_note-FOOTNOTEApostol1976§3.9-12">[10]</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)}</annotation>

</semantics>

</math><img src="./6412d2052c8e54a65d596a419b429d00fd18d53b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.606ex; height:2.843ex;" alt="{\displaystyle \mu (n)}" loading="lazy"> sections</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)=0}</annotation>

</semantics>

</math><img src="./d96dd239de31328b7f9c1a03dd7a8dd1a5bdf710.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.867ex; height:2.843ex;" alt="{\displaystyle \mu (n)=0}" loading="lazy"> <a href="If_and_only_if" title="If and only if">if and only if</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is divisible by the square of a prime. The first numbers with this property are

<dl><dd>4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, ... (sequence <a href="https://oeis.org/A013929" class="extiw external" title="oeis:A013929">A013929</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is prime, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (n)=-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)=-1}</annotation>

</semantics>

</math><img src="./1d5101b3cddb8789ba1b0bccdefd1014a3c6b297.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.675ex; height:2.843ex;" alt="{\displaystyle \mu (n)=-1}" loading="lazy">, but the converse is not true. The first non prime <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (n)=-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)=-1}</annotation>

</semantics>

</math><img src="./1d5101b3cddb8789ba1b0bccdefd1014a3c6b297.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.675ex; height:2.843ex;" alt="{\displaystyle \mu (n)=-1}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 30=2\times 3\times 5}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 30=2\times 3\times 5}</annotation>

</semantics>

</math><img src="./aefe3cdb2ce48ab09def8d3cb509bb01fa37769f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.592ex; height:2.176ex;" alt="{\displaystyle 30=2\times 3\times 5}" loading="lazy">. The first such numbers with three distinct prime factors (<a href="Sphenic_number" title="Sphenic number">sphenic numbers</a>) are

<dl><dd>30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, ... (sequence <a href="https://oeis.org/A007304" class="extiw external" title="oeis:A007304">A007304</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl>

and the first such numbers with 5 distinct prime factors are

<dl><dd>2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7770, 7854, 8610, 8778, 8970, 9030, 9282, 9570, 9690, ... (sequence <a href="https://oeis.org/A046387" class="extiw external" title="oeis:A046387">A046387</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Mertens_function">Mertens function</h2></div>

In number theory another <a href="Arithmetic_function" title="Arithmetic function">arithmetic function</a> closely related to the Möbius function is the <a href="Mertens_function" title="Mertens function">Mertens function</a>, defined by

</mrow>

</mrow>

</munderover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(n)=\sum _{k=1}^{n}\mu (k)}</annotation>

</semantics>

</math><img src="./4728d26512fe7f58647568fe9a6d3e8a5d66d018.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.909ex; height:6.843ex;" alt="{\displaystyle M(n)=\sum _{k=1}^{n}\mu (k)}" loading="lazy"></dd></dl>

for every natural number n. This function is closely linked with the positions of zeroes of the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>. See the article on the <a href="Mertens_conjecture" title="Mertens conjecture">Mertens conjecture</a> for more information about the connection between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(n)}</annotation>

</semantics>

</math><img src="./9c856803491e153263b91aacc660e26ef129d467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.646ex; height:2.843ex;" alt="{\displaystyle M(n)}" loading="lazy"> and the <a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>.

From the formula

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</munder>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)=\sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}e^{2\pi i{\frac {k}{n}}},}</annotation>

</semantics>

</math><img src="./5b4102f2bc73536bbb65fe03e176a45b68a8d56f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:22.168ex; height:8.176ex;" alt="{\displaystyle \mu (n)=\sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}e^{2\pi i{\frac {k}{n}}},}" loading="lazy"></dd></dl>

it follows that the Mertens function is given by

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(n)=-1+\sum _{a\in {\mathcal {F}}_{n}}e^{2\pi ia},}</annotation>

</semantics>

</math><img src="./12d92e66102c8a84cfc12143be41cd845d2b6d63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.219ex; height:5.843ex;" alt="{\displaystyle M(n)=-1+\sum _{a\in {\mathcal {F}}_{n}}e^{2\pi ia},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{n}}</annotation>

</semantics>

</math><img src="./1aedfc87ef7b181d03cfd9b4d74662cdea74e5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.89ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{n}}" loading="lazy"> is the <a href="Farey_sequence" title="Farey sequence">Farey sequence</a> of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">.

This formula is used in the proof of the <a href="Farey_sequence#Riemann_hypothesis" title="Farey sequence">Franel–Landau theorem</a>.<a href="#cite_note-FOOTNOTEEdwards1974Ch._12.2-13">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Incidence_algebras">Incidence algebras</h3></div>

In <a href="Combinatorics" title="Combinatorics">combinatorics</a>, every locally finite <a href="Partially_ordered_set" title="Partially ordered set">partially ordered set</a> (poset) is assigned an <a href="Incidence_algebra" title="Incidence algebra">incidence algebra</a>. One distinguished member of this algebra is that poset's "Möbius function". The classical Möbius function treated in this article is essentially equal to the Möbius function of the set of all positive integers partially ordered by <a href="Divisor" title="Divisor">divisibility</a>. See the article on <a href="Incidence_algebra" title="Incidence algebra">incidence algebras</a> for the precise definition and several examples of these general Möbius functions.

<div class="mw-heading mw-heading3"><h3 id="Popovici's_function">Popovici's function</h3></div>

Constantin Popovici<a href="#cite_note-FOOTNOTEPopovici1963493–499-14">[12]</a> defined a generalised Möbius function

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{k}=\mu *\cdots *\mu }</annotation>

</semantics>

</math><img src="./9d9dfd18657bfadcbeba369369ef300e965b7840.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.505ex; height:2.176ex;" alt="{\displaystyle \mu _{k}=\mu *\cdots *\mu }" loading="lazy"> to be the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-fold <a href="Dirichlet_convolution" title="Dirichlet convolution">Dirichlet convolution</a> of the Möbius function with itself. It is thus again a multiplicative function with

<msub>

</mrow>

</msub>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{k}\left(p^{a}\right)=(-1)^{a}{\binom {k}{a}}\ }</annotation>

</semantics>

</math><img src="./70219a3a9902915491f9d2d77e56fc398cb00754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.17ex; height:6.176ex;" alt="{\displaystyle \mu _{k}\left(p^{a}\right)=(-1)^{a}{\binom {k}{a}}\ }" loading="lazy"></dd></dl>

where the <a href="Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a> is taken to be zero if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a>k}</annotation>

</semantics>

</math><img src="./daf927ffe6370d760125d2270b3912c9626df7fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.54ex; height:2.176ex;" alt="{\displaystyle a>k}" loading="lazy">. The definition may be extended to complex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> by reading the binomial as a polynomial in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">.<a href="#cite_note-FOOTNOTESándorCrstici2004107-15">[13]</a>

<div class="mw-heading mw-heading2"><h2 id="Implementations">Implementations</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://functions.wolfram.com/NumberTheoryFunctions/MoebiusMu/">Mathematica</a></li>

<li><a rel="nofollow" class="external text" href="https://maxima.sourceforge.io/docs/manual/maxima_singlepage.html#index-moebius">Maxima</a></li>

<li><a rel="nofollow" class="external text" href="https://www.geeksforgeeks.org/program-mobius-function/">geeksforgeeks</a> C++, Python3, Java, C#, PHP, JavaScript</li>

<li><a rel="nofollow" class="external text" href="https://rosettacode.org/wiki/M%C3%B6bius_function">Rosetta Code</a></li>

/* start https://en.wikipedia.org/ */

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</style><div class="div-col">

<ul><li><a href="Liouville_function" title="Liouville function">Liouville function</a></li>

<li><a href="Mertens_function" title="Mertens function">Mertens function</a></li>

<li><a href="Ramanujan's_sum" title="Ramanujan's sum">Ramanujan's sum</a></li>

<li><a href="Sphenic_number" title="Sphenic number">Sphenic number</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> Hardy & Wright, Notes on ch. XVI: "... <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)}</annotation>

</semantics>

</math><img src="./6412d2052c8e54a65d596a419b429d00fd18d53b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.606ex; height:2.843ex;" alt="{\displaystyle \mu (n)}" loading="lazy"> occurs implicitly in the works of Euler as early as 1748, but Möbius, in 1832, was the first to investigate its properties systematically". (<a href="#CITEREFHardyWright1980">Hardy & Wright 1980</a>, Notes on ch. XVI)

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> In the <a href="Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a> (1801) <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> showed that the sum of the primitive roots (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mod p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mod p}</annotation>

</semantics>

</math><img src="./89fbffea5527c156da16f8402f3fde406d320288.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.044ex; height:2.509ex;" alt="{\displaystyle \mod p}" loading="lazy">) is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (p-1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (p-1)}</annotation>

</semantics>

</math><img src="./d8c88cd31f74cb2a5c4ce8a088a3e32bf82ca5af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.383ex; height:2.843ex;" alt="{\displaystyle \mu (p-1)}" loading="lazy">, (see <a href="#Properties_and_applications">#Properties and applications</a>) but he didn't make further use of the function. In particular, he didn't use Möbius inversion in the Disquisitiones.<a href="#cite_note-FOOTNOTEGauss1986Art._81-2">[1]</a> The <a href="Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a> has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

<li id="cite_note-FOOTNOTEGauss1986Art._81-2">^ <a href="#cite_ref-FOOTNOTEGauss1986Art._81_2-0">a</a> <a href="#cite_ref-FOOTNOTEGauss1986Art._81_2-1">b</a> <a href="#CITEREFGauss1986">Gauss 1986</a>, Art. 81.

</li>

<li id="cite_note-FOOTNOTEMöbius1832105–123-4"><a href="#cite_ref-FOOTNOTEMöbius1832105–123_4-0">^</a> <a href="#CITEREFMöbius1832">Möbius 1832</a>, pp. 105–123.

</li>

<li id="cite_note-FOOTNOTEAbramowitzStegun1972826-5"><a href="#cite_ref-FOOTNOTEAbramowitzStegun1972826_5-0">^</a> <a href="#CITEREFAbramowitzStegun1972">Abramowitz & Stegun 1972</a>, p. 826.

</li>

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</style><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/MoebiusFunction.html">"Möbius Function"</a>. mathworld.wolfram.com. Retrieved 1 October 2024.</cite>

</li>

<li id="cite_note-FOOTNOTEJacobson2009§4.13-7"><a href="#cite_ref-FOOTNOTEJacobson2009§4.13_7-0">^</a> <a href="#CITEREFJacobson2009">Jacobson 2009</a>, §4.13.

</li>

<li id="cite_note-FOOTNOTEBostConnes1995411–457-8"><a href="#cite_ref-FOOTNOTEBostConnes1995411–457_8-0">^</a> <a href="#CITEREFBostConnes1995">Bost & Connes 1995</a>, pp. 411–457.

</li>

<li id="cite_note-FOOTNOTEHardyWright1980(16.6.4),_p._239-9"><a href="#cite_ref-FOOTNOTEHardyWright1980(16.6.4),_p._239_9-0">^</a> <a href="#CITEREFHardyWright1980">Hardy & Wright 1980</a>, (16.6.4), p. 239.

</li>

<li id="cite_note-FOOTNOTEApostol1976-10"><a href="#cite_ref-FOOTNOTEApostol1976_10-0">^</a> <a href="#CITEREFApostol1976">Apostol 1976</a>.

</li>

<li id="cite_note-FOOTNOTEKline2020-11"><a href="#cite_ref-FOOTNOTEKline2020_11-0">^</a> <a href="#CITEREFKline2020">Kline 2020</a>.

</li>

<li id="cite_note-FOOTNOTEApostol1976§3.9-12"><a href="#cite_ref-FOOTNOTEApostol1976§3.9_12-0">^</a> <a href="#CITEREFApostol1976">Apostol 1976</a>, §3.9.

</li>

<li id="cite_note-FOOTNOTEEdwards1974Ch._12.2-13"><a href="#cite_ref-FOOTNOTEEdwards1974Ch._12.2_13-0">^</a> <a href="#CITEREFEdwards1974">Edwards 1974</a>, Ch. 12.2.

</li>

<li id="cite_note-FOOTNOTEPopovici1963493–499-14"><a href="#cite_ref-FOOTNOTEPopovici1963493–499_14-0">^</a> <a href="#CITEREFPopovici1963">Popovici 1963</a>, pp. 493–499.

</li>

<li id="cite_note-FOOTNOTESándorCrstici2004107-15"><a href="#cite_ref-FOOTNOTESándorCrstici2004107_15-0">^</a> <a href="#CITEREFSándorCrstici2004">Sándor & Crstici 2004</a>, p. 107.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2></div>

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<ul><li><cite id="CITEREFAbramowitzStegun1972" class="citation book cs1">Abramowitz, Milton; Stegun, Irene A. (1972) [1964]. Handbook of mathematical functions: with formulas, graphs and mathematical tables [conference under the auspices of the National science foundation and the Massachusetts institute of technology]. Dover books on advanced mathematics. New York: Dover. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-61272-0</bdi>.</cite></li>

<li><cite id="CITEREFApostol1976" class="citation book cs1"><a href="Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (1976). Introduction to analytic number theory. Undergraduate Texts in Mathematics. New York; Heidelberg: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90163-3</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929">0434929</a>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0335.10001">0335.10001</a>.</cite></li>

<li><cite id="CITEREFBostConnes1995" class="citation journal cs1">Bost, J.-B.; Connes, Alain (1995). <a rel="nofollow" class="external text" href="https://cds.cern.ch/record/283504">"Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory"</a>. Selecta Mathematica. New Series. 1 (3): 411–457. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01589495">10.1007/BF01589495</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:116418599">116418599</a>.</cite></li>

<li><cite id="CITEREFDelégliseRivat1996" class="citation journal cs1">Deléglise, Marc; Rivat, Joël (1996). <a rel="nofollow" class="external text" href="https://projecteuclid.org/euclid.em/1047565447">"Computing the summation of the Möbius function"</a>. Experimental Mathematics. 5 (4): 291–295. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F10586458.1996.10504594">10.1080/10586458.1996.10504594</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:574146">574146</a>.</cite></li>

<li><cite id="CITEREFEdwards1974" class="citation book cs1"><a href="Harold_Edwards_(mathematician)" title="Harold Edwards (mathematician)">Edwards, Harold</a> (1974). Riemann's Zeta Function. Mineola, New York: Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-41740-9</bdi>.</cite></li>

<li><cite id="CITEREFGauss1965" class="citation book cs1"><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss, Carl Friedrich</a> (1965). Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory). Translated by Maser, H. (2nd ed.). New York: Chelsea. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8284-0191-8</bdi>.</cite></li>

<li><cite id="CITEREFGauss1986" class="citation book cs1"><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss, Carl Friedrich</a> (1986). Disquisitiones Arithemeticae. Translated by Clarke, Arthur A. (corrected 2nd ed.). New York: <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-96254-9</bdi>.</cite></li>

<li><cite id="CITEREFHardyWright1980" class="citation book cs1"><a href="G._H._Hardy" title="G. H. Hardy">Hardy, G. H.</a>; <a href="E._M._Wright" title="E. M. Wright">Wright, E. M.</a> (1980) [First edition published 1938]. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth00hard">An Introduction to the Theory of Numbers</a> (5th ed.). Oxford: <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-853171-5</bdi> – via <a href="Internet_Archive" title="Internet Archive">Internet Archive</a>.</cite></li>

<li><cite id="CITEREFKline2020" class="citation journal cs1">Kline, Jeffery (2020). <a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Kline/kline4.pdf">"Unital Sums of the Möbius and Mertens Functions"</a> (PDF). Journal of Integer Sequences. 23 (8): 1–17.</cite></li>

<li><cite id="CITEREFJacobson2009" class="citation book cs1"><a href="Nathan_Jacobson" title="Nathan Jacobson">Jacobson, Nathan</a> (2009) [First published 1985]. Basic algebra I (2nd ed.). Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-47189-1</bdi>.</cite></li>

<li><cite id="CITEREFKlimov2001" class="citation cs2">Klimov, N. I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Möbius_function">"Möbius function"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li><cite id="CITEREFMöbius1832" class="citation journal cs1"><a href="August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">Möbius, A. F.</a> (1832). <a rel="nofollow" class="external text" href="https://www.digizeitschriften.de/en/dms/img/?PID=GDZPPN002138654">"Über eine besondere Art von Umkehrung der Reihen"</a>. <a href="Crelle's_Journal" title="Crelle's Journal">Journal für die reine und angewandte Mathematik</a>. 9: 105–123.</cite></li>

<li><cite id="CITEREFPegg2003" class="citation web cs2"><a href="Ed_Pegg_Jr." title="Ed Pegg Jr.">Pegg, Ed Jr</a> (2003), <a rel="nofollow" class="external text" href="http://www.mathpuzzle.com/MAA/02-Mobius%20Function/mathgames_11_03_03.html">"The Möbius function (and squarefree numbers)"</a>, Ed Pegg's Math Games</cite></li>

<li><cite id="CITEREFPopovici1963" class="citation journal cs1">Popovici, Constantin P. (1963). "A generalization of the Möbius function". Studii şi Cercetări Matematice. 14: 493–499. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0181602">0181602</a>.</cite></li>

<li><cite id="CITEREFSándorCrstici2004" class="citation book cs1">Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-4020-2546-7</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1079.11001">1079.11001</a>.</cite></li>

<li><cite id="CITEREFSándorMitrinovićCrstici2006" class="citation book cs1">Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. pp. 187–226. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-4020-4215-9</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1151.11300">1151.11300</a>.</cite></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/MoebiusFunction.html">"Möbius function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li></ul></div><div><div class="zim-footer">

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