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:
- $\forall m, n \in K: f \left({m n}\right) = f \left({m}\right) f \left({n}\right)$
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.
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- It can easily be proved by induction that $\forall k \in \N: \left({f \left({n}\right)}\right)^k = f \left({n^k}\right)$ 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.
