Questions on the Möbius function μ(n), an arithmetic function used in number theory.

The Möbius function $\mu(n)$ is defined for positive integers $n$ using their prime factorization. We have $\mu(1)=1$ and if $n>2$, $n=p_1^{a_1}\cdots p_r^{a_r}$, then $$\mu(n)=\begin{cases} 0 &\text{if }n\text{ is not square-free, that is, }a_k>1\text{ for some }k\\ (-1)^r &\text{otherwise.} \end{cases}$$

The Möbius function is multiplicative: if $m$ and $n$ are relatively prime, then

$$\mu(mn) = \mu(m) \mu(n).$$

One particularly important use of the Möbius function is the Möbius-inversion formula, which states that $\mu$ is the Dirichlet inverse of the constant function $1$.