\begin{definition}[Definition:Möbius Function]
Let $n \in \Z_{>0}$, that is, a strictly positive integer.
The '''Möbius function''' is the function $\mu: \Z_{>0} \to \Z_{>0}$ defined as:
:$\map \mu n = \begin{cases}
1 & : n = 1 \\
0 & : \exists p \in \mathbb P: p^2 \divides n\\
\left({-1}\right)^k & : n = p_1 p_2 \ldots p_k: p_i \in \mathbb P
\end{cases}$
That is:
:$\map \mu n = 1$ if $n = 1$
:$\map \mu n = 0$ if $n$ has any divisor which is the square of a prime, that is $n$ is not square-free
:$\map \mu n = \paren {-1}^k$ if $n$ has $k$ distinct prime divisors.
\end{definition}