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

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.