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

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