## Definition

Let $R$ be a unique factorization domain.

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

Then $f$ is additive if and only if:

For all coprime $x,y\in R$, $f(xy) = f(x) + f(y)$

### Arithmetic Function

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

Then $f$ is additive if and only if:

$m \perp n \implies \map f {m n} = \map f m + \map f n$