Dirichlet Convolution Preserves Multiplicativity

Theorem
Let $f, g: \N \to \C$ be multiplicative arithmetic functions.

Then their Dirichlet convolution $f * g$ is again multiplicative.

Proof
Let $m, n$ be coprime integers.

By definition of multiplicative functions, we have:


 * $(1): \quad \map f {m n} = \map f m \map f n$
 * $(2): \quad \map g {m n} = \map g m \map g n$

Also see

 * Properties of Dirichlet Convolution