# 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:

- $\forall m, n \in K: f \left({m n}\right) = f \left({m}\right) f \left({n}\right)$

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 $\forall k \in \N: \left({f \left({n}\right)}\right)^k = f \left({n^k}\right)$ if and only if $f$ is completely multiplicative.

- Completely Multiplicative Function is Multiplicative, but not necessarily the other way about.

- Results about
**completely multiplicative functions**can be found here.