Möbius function

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

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

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

Möbius inversion formula

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

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

corresponds to

$G(s)=F(s)\zeta (s).\,$

We therefore have

$F(s)=G(s)M(s),\,$,

giving the Möbius inversion formula

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

A useful special case is the formula

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

Mertens conjecture

The Mertens conjecture is that the summatory function

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

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