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 $M(s)=1/\zeta (s)$.

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.