Dirichlet Convolution Preserves Multiplicativity

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

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

Proof
Let $m, n$ be coprime integers.

By definition of multiplicative functions, we have:


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

Also see

 * Properties of Dirichlet Convolution