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 $F(s)$ and $G(s)$ converge absolutely.

Then $F(s) \cdot G(s) = 1$.

Proof
Let $\varepsilon$ be the identity arithmetic function.

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

By Dirichlet Series of Convolution of Arithmetic Functions, $F(s)G(s) = 1$.