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The Möbius function is an
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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.
[[Arithmetic function|arithmetic function]] of a natural 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
 
The Möbius function is a
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[[Distribution of prime numbers|distribution of prime numbers]] in the natural series.
 
[[Distribution of prime numbers|distribution of prime numbers]] in the natural series.
  
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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
  
The Möbius function is related to the Riemann zeros via the explicit formula
 
  
\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}
 
  
Where g(x) and h(x) form a Fourier transform pair and the sum on the right is over the imaginary part of the nontrivial zeros.
 
  
 
====Comments====
 
====Comments====
 
The multiplicative arithmetic functions form a
 
The multiplicative arithmetic functions form a
[[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. e.g.
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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]].
[[Möbius series|Möbius series]].
 
  
 
====References====
 
====References====
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|valign="top"|{{Ref|Wa}}||valign="top"|  A. Walfisz,  "Weylsche Exponentialsummen in der neueren Zahlentheorie", Deutsch. Verlag Wissenschaft.  (1963)  {{MR|0220685}}   
 
|valign="top"|{{Ref|Wa}}||valign="top"|  A. Walfisz,  "Weylsche Exponentialsummen in der neueren Zahlentheorie", Deutsch. Verlag Wissenschaft.  (1963)  {{MR|0220685}}   
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====Comments====
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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.
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====References====
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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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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=30443
This article was adapted from an original article by N.I. Klimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article