# Definition:Completely Multiplicative Function

## Definition

Let $K$ be a field.

Let $f: K \to K$ be a function on $K$.

Then $f$ is described as completely multiplicative if and only if:

$\forall m, n \in K: \map f {m n} = \map f m \map f n$

That is, a completely multiplicative function is one where the value of a product of two numbers equals the product of the value of each one individually.

## Also see

• It can easily be proved by induction that $\forall k \in \N: \paren {\map f n}^k = \map f {n^k}$ if and only if $f$ is completely multiplicative.
• Results about completely multiplicative functions can be found here.