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

This page or section has statements made on it that ought to be extracted and proved in a Theorem page. In particular: There are results in here which need extracting You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed. To discuss this page in more detail, feel free to use the talk page.

• 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.

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

• Results about completely multiplicative functions can be found here.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): arithmetic function