Möbius function

The Möbius function of number theory is the function $\mu:\mathbb{Z}^+\to\{-1,0,1\}$ defined by

In other words, $\mu (n) = 0$ if $n$ is not a square-free integer, while $\mu (n) = (-1)^r$ if $n$ is square-free with $r$ prime factors. The function $\mu$ is a multiplicative function, and obeys the identity