The Möbius function of number theory is the function $\mu:\mathbb{Z}^+\to\{-1,0,1\}$ defined by $$ \mu (n) = \begin{cases} 1, &\text{if $n=1$}\\ 0, &\text{if $p^2 | n$ for some prime $p$} \\ (-1)^r, &\text{if $n = p_1 p_2 \cdots p_r$, where the $p_i$ are distinct primes.} \end{cases} $$

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 $$ \sum_{d | n} \mu(d) = \begin{cases} 1 & \text{if $n = 1$}\\ 0 & \text{if $n > 1$} \end{cases} $$ where $d$ runs through the positive divisors of $n$