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


 * $$\sum_{n\le x} \mu(n) \le \sqrt{x} .\,$$

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