Definition:Additive Function on UFD

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 $x, y \in R$: $\map f {x y} = \map f x + \map f y$

Arithmetic Function

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

Then $f$ is additive if and only if:

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

Also see

• Definition:Multiplicative Function on UFD

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