Dirichlet Series of Inverse of Arithmetic Function

Theorem

Let $f : \N \to\C$ be an arithmetic function.

Let $g : \N \to \C$ be an Dirichlet inverse of $f$.

Let $F, G$ be their Dirichlet series.

Let $s \in \C$ such that both $\map F s$ and $\map G s$ converge absolutely.

Then $\map F s \cdot \map G s = 1$.

Proof

Let $\varepsilon$ be the identity arithmetic function.

By Dirichlet Series of Identity Arithmetic Function, $\varepsilon$ has Dirichlet series $\map E s = 1$.

By Dirichlet Series of Convolution of Arithmetic Functions, $\map F s \map G s = 1$.

$\blacksquare$