Multiplicativity of the Möbius function

Statement

The Möbius function $\mu$ is a multiplicative function. That is, $\mu(1) = 1$ and $\mu(mn) = \mu(m) \mu(n)$ for all coprime positive integers $m$ and $n$.

Remarks

The function $\mu$ is not completely multiplicative: $\mu(4) = 0$, but $\mu(2)^2 = 1$. The proof: for coprime $m$ and $n$, the product $mn$ is squarefree exactly when $m$ and $n$ are both squarefree. Then $\omega(mn) = \omega(m) + \omega(n)$.

Explanations

L1 human

For example, $\mu(5) = -1$ and $\mu(6) = 1$, and $\mu(30) = -1 = \mu(5) \mu(6)$. When $m$ and $n$ have no common prime, their prime factors just combine, so the signs multiply.