# Definition:Completely Multiplicative Function

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## 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**.

- 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): Entry:**arithmetic function**