Dirichlet Series of Convolution of Arithmetic Functions

Theorem
Let $f,g : \N \to\C$ be arithmetic functions.

Let $h = f*g$ be their Dirichlet convolution.

Let $F,G,H$ be their Dirichlet series.

Let $s$ be a complex number such that $F(s)$ and $G(s)$ converge absolutely.

Then $H(s)$ converges absolutely to $F(s)G(s)$.

Also see

 * Dirichlet Series of Inverse of Arithmetic Function
 * Upper Bound for Abscissa of Absolute Convergence of Product of Dirichlet Series