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An [[Arithmetic function|arithmetic function]] of natural argument: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642801.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642802.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642803.png" /> is divisible by the square of a prime number, otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642804.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642805.png" /> is the number of prime factors of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642806.png" />. This function was introduced by A. Möbius in 1832.
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The Möbius function is an [[arithmetic function]] of a natural number argument $n$ with $\mu(1)=1$, $\mu(n)=0$ if $n$ is divisible by the square of a prime number, otherwise $\mu(n) = (-1)^k$, where $k$ is the number of prime factors of $n$. This function was introduced by A. Möbius in 1832.
  
The Möbius function is a [[Multiplicative arithmetic function|multiplicative arithmetic function]]; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642807.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642808.png" />. It is used in the study of other arithmetic functions; it appears in inversion formulas (see, e.g. [[Möbius series|Möbius series]]). The following estimate is known for the mean value of the Möbius function [[#References|[2]]]:
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The Möbius function is a
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[[Multiplicative arithmetic function|multiplicative arithmetic function]]; $\sum_{d|n}\mu(d) = 0$ if $n>1$. It is used in the study of other arithmetic functions; it appears in inversion formulas (see, e.g.
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[[Möbius series|Möbius series]]). The following estimate is known for the mean value of the Möbius function
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{{Cite|Wa}}:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642809.png" /></td> </tr></table>
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$${1\over x}\Big|\sum_{n\le x}\mu(n)\Big| \le \exp\{-c \ln^{3/5} x(\ln\ln x)^{-1/5} \},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m06428010.png" /> is a constant. The fact that the mean value tends to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m06428011.png" /> implies an asymptotic law for the [[Distribution of prime numbers|distribution of prime numbers]] in the natural series.
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where $c$ is a constant. The fact that the mean value tends to zero as $x\to \infty$ implies an asymptotic law for the
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[[Distribution of prime numbers|distribution of prime numbers]] in the natural series.
  
The Möbius function is related to the Riemann zeros via the formula
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The Möbius function satsifies the explicit formula
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$$ \sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}g(\log n)=\sum_{\rho}\frac{h( \gamma)}{\zeta '( \rho )}+\sum_{n=1}^{\infty} \frac{1}{\zeta ' (-2n)} \int_{-\infty}^{\infty}dxg(x)e^{-(2n+1/2)x} $$
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Where $$ g(u)= \frac{1}{2\pi} \int_{-\infty}^\infty h(x)\exp(-iux) $$
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form a Fourier transformation pair
  
\begin{equation} \sum_{n=1}^{\infty}\frac{\mu(n)}{\sqrt{n}} g \log n = \sum_t \frac{h(t)}{\zeta'(1/2+it)}+2\sum_{n=1}^\infty \frac{ (-1)^{n} (2\pi )^{2n}}{(2n)! \zeta(2n+1)}\int_{-\infty}^{\infty}g(x) e^{-x(2n+1/2)} \, dx,\end{equation}
 
  
Here g(x) and h(x) form a [[Fourier transform]] pair.
 
  
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,  "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Walfisz,  "Weylsche Exponentialsummen in der neueren Zahlentheorie" , Deutsch. Verlag Wissenschaft.  (1963)</TD></TR></table>
 
  
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====Comments====
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The multiplicative arithmetic functions form a
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[[Group|group]] under the convolution product $(f*g)(n) = \sum_{d|n}f(d)g(n/d)$. The Möbius function is in fact the inverse of the constant multiplicative function $E$ (defined by $E(n)=1$ for all $n\in \N$) under this convolution product. From this there follows many  "inversion formulas" , cf. [[Möbius inversion]].
  
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====References====
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{|
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|-
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|valign="top"|{{Ref|HaWr}}||valign="top"|  G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers", Clarendon Press  (1979)  {{MR|0568909}} 
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|-
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|valign="top"|{{Ref|Vi}}||valign="top"|  I.M. Vinogradov,  "Elements of number theory", Dover, reprint  (1954)  (Translated from Russian)  {{MR|0062138}} 
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|-
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|valign="top"|{{Ref|Wa}}||valign="top"|  A. Walfisz,  "Weylsche Exponentialsummen in der neueren Zahlentheorie", Deutsch. Verlag Wissenschaft.  (1963)  {{MR|0220685}} 
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|-
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|}
  
 
====Comments====
 
====Comments====
The multiplicative arithmetic functions form a [[Group|group]] under the convolution product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m06428012.png" />. The Möbius function is in fact the inverse of the constant multiplicative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m06428013.png" /> (defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m06428014.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m06428015.png" />) under this convolution product. From this there follows many  "inversion formulas" , cf. e.g. [[Möbius series|Möbius series]].
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For the Möbius function associated to a [[partially ordered set]], see [[Enumeration theory]].  In this context, the arithmetic Möbius function defined in this article appears as the function associated to the natural numbers ordered by divisibility.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Clarendon Press (1979)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|KRY}}||valign="top"| Kung, Joseph P. S.; Rota, Gian-Carlo; Yan, Catherine H. ''Combinatorics. The Rota way''. Cambridge University Press (2009) ISBN 978-0-521-73794-4 {{ZBL|1159.05002}}
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|-
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|}

Revision as of 11:54, 17 January 2020

2020 Mathematics Subject Classification: Primary: 11A [MSN][ZBL]

The Möbius function is an arithmetic function of a natural number argument $n$ with $\mu(1)=1$, $\mu(n)=0$ if $n$ is divisible by the square of a prime number, otherwise $\mu(n) = (-1)^k$, where $k$ is the number of prime factors of $n$. This function was introduced by A. Möbius in 1832.

The Möbius function is a multiplicative arithmetic function; $\sum_{d|n}\mu(d) = 0$ if $n>1$. It is used in the study of other arithmetic functions; it appears in inversion formulas (see, e.g. Möbius series). The following estimate is known for the mean value of the Möbius function [Wa]:

$${1\over x}\Big|\sum_{n\le x}\mu(n)\Big| \le \exp\{-c \ln^{3/5} x(\ln\ln x)^{-1/5} \},$$

where $c$ is a constant. The fact that the mean value tends to zero as $x\to \infty$ implies an asymptotic law for the distribution of prime numbers in the natural series.

The Möbius function satsifies the explicit formula

$$ \sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}g(\log n)=\sum_{\rho}\frac{h( \gamma)}{\zeta '( \rho )}+\sum_{n=1}^{\infty} \frac{1}{\zeta ' (-2n)} \int_{-\infty}^{\infty}dxg(x)e^{-(2n+1/2)x} $$

Where $$ g(u)= \frac{1}{2\pi} \int_{-\infty}^\infty h(x)\exp(-iux) $$

form a Fourier transformation pair



Comments

The multiplicative arithmetic functions form a group under the convolution product $(f*g)(n) = \sum_{d|n}f(d)g(n/d)$. The Möbius function is in fact the inverse of the constant multiplicative function $E$ (defined by $E(n)=1$ for all $n\in \N$) under this convolution product. From this there follows many "inversion formulas" , cf. Möbius inversion.

References

[HaWr] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers", Clarendon Press (1979) MR0568909
[Vi] I.M. Vinogradov, "Elements of number theory", Dover, reprint (1954) (Translated from Russian) MR0062138
[Wa] A. Walfisz, "Weylsche Exponentialsummen in der neueren Zahlentheorie", Deutsch. Verlag Wissenschaft. (1963) MR0220685

Comments

For the Möbius function associated to a partially ordered set, see Enumeration theory. In this context, the arithmetic Möbius function defined in this article appears as the function associated to the natural numbers ordered by divisibility.

References

[KRY] Kung, Joseph P. S.; Rota, Gian-Carlo; Yan, Catherine H. Combinatorics. The Rota way. Cambridge University Press (2009) ISBN 978-0-521-73794-4 Zbl 1159.05002
How to Cite This Entry:
Möbius function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=M%C3%B6bius_function&oldid=29874
This article was adapted from an original article by N.I. Klimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article