Completely multiplicative function

Definition

An arithmetic function $f$ is completely multiplicative if $f(1) = 1$ and $f(mn) = f(m) f(n)$ for all positive integers $m$ and $n$.

Remarks

Each completely multiplicative function is multiplicative. The converse fails: Euler's totient function has $\varphi(4) = 2$ but $\varphi(2)^2 = 1$. A completely multiplicative function is determined by its values at the primes. Examples are $n \mapsto n^k$, the Liouville function and, for a fixed odd prime $p$, the Legendre symbol $n \mapsto \left(\frac{n}{p}\right)$.

Explanations

L1 human

A completely multiplicative function respects every product, not only products of coprime numbers. So it is enough to know its values at the primes, as for the function $n \mapsto n^2$.