Definition:Multiplicative Function on UFD

Definition

Let $R$ be a unique factorization domain.

Let $f : R \to \C$ be a complex-valued function.

Then $f$ is multiplicative if and only if:

For all coprime $x, y\in R$: $f \left({x y}\right) = f \left({x}\right) f \left({y}\right)$

Arithmetic Function

Let $f : \N \to \C$ be an arithmetic function.

Then $f$ is multiplicative if and only if:

$m \perp n \implies f \left({m n}\right) = f \left({m}\right) f \left({n}\right)$

where $m \perp n$ denotes that $m$ is coprime to $n$.