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
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- 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