Dirichlet Convolution Preserves Multiplicativity/General Result

Theorem

Let $f, g$ be multiplicative functions.

Let $S \subset \N$ be a set of natural numbers with the property:

$m n \in S, \map \gcd {m, n} = 1 \implies m, n \in S$

Define:

$\map {\paren {f*_S g} } n = \ds \sum_{\substack {d \mathop \divides n \\ d \mathop \in S} } \map f d \map g {n / d}$


Then $f*_S g$ is multiplicative.


Proof


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