Möbius function

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 satisfies 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.

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

[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
[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