Difference between revisions of "Möbius function"
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| − | The Möbius function is an | + | 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. |
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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==== | ||
| + | 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 16:50, 20 December 2015
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.
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 |
Möbius function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=M%C3%B6bius_function&oldid=34168