Definition:Completely Multiplicative Function

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.