# Möbius function

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In number theory, the Möbius function μ(n) is an arithmetic function which takes the values -1, 0 or +1 depending on the prime factorisation of its input n.

If the positive integer n has a repeated prime factor then μ(n) is defined to be zero. If n is square-free, then μ(n) = +1 if n has an even number of prime factors and -1 if n has an odd number of prime factors.

The Möbius function is multiplicative, and hence the associated formal Dirichlet series has an Euler product

${\displaystyle M(s)=\sum _{n}\mu (n)n^{-s}=\prod _{p}\left(1-p^{-s}\right).\,}$

Comparison with the zeta function shows that formally at least ${\displaystyle M(s)=1/\zeta (s)}$.

## Möbius inversion formula

Let f be an arithmetic function and F(s) the corresponding formal Dirichlet series. The Dirichlet convolution

${\displaystyle g(n)=\sum _{d|n}f(d)\,}$

corresponds to

${\displaystyle G(s)=F(s)\zeta (s).\,}$

We therefore have

${\displaystyle F(s)=G(s)M(s),\,}$,

giving the Möbius inversion formula

${\displaystyle f(n)=\sum _{d|n}\mu (d)g(n/d).\,}$

A useful special case is the formula

${\displaystyle \sum _{d|n}\mu (d)=1{\mbox{ if }}n=1{\mbox{ and }}0{\mbox{ if }}n>1.\,}$

## Mertens conjecture

The Mertens conjecture is that the summatory function

${\displaystyle \sum _{n\leq x}\mu (n)\leq {\sqrt {x}}.\,}$

The truth of the Mertens conjecture would imply the Riemann hypothesis. However, computations by Andrew Odlyzko have shown that the Mertens conjecture is false.