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 iff:

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

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.

It can easily be proved by induction that $\forall k \in \N: \left({f \left({n}\right)}\right)^k = f \left({n^k}\right)$ iff $f$ is completely multiplicative.

It follows trivially that a completely multiplicative function is also a multiplicative function, but not necessarily the other way about.