Definition:Multiplicative Function on UFD
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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:
- $m \perp n \implies \map f {m n} = \map f m \map f n$
where $m \perp n$ denotes that $m$ is coprime to $n$.