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:
 * $m n \in S, \; \gcd \left({m, n}\right) = 1 \implies m, n \in S$

Define:
 * $\left({f*_S g}\right) \left({n}\right) = \displaystyle \sum_{\substack {d \mathop \backslash n \\ d \mathop \in S} } f \left({d}\right) g \left({n / d}\right)$

Then $f*_S g$ is multiplicative.