Upper Bound for Abscissa of Absolute Convergence of Product of Dirichlet Series

Theorem
Let $f, g: \N \to \C$ be arithmetic functions with Dirichlet convolution $h = f * g$.

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

Let $\sigma_f, \sigma_g, \sigma_h$ be their abscissae of absolute convergence.

Then:
 * $\sigma_h \le \max \set {\sigma_f, \sigma_g}$

Proof
Follows from Dirichlet Series of Convolution of Arithmetic Functions