Möbius function

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

The Möbius function is an 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 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 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

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. e.g. Möbius series.

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