Moebius 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


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

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