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 $\sigma_h \leq \max(\sigma_f, \sigma_g)$.

Proof
Follows from Dirichlet Series of Convolution of Arithmetic Functions