Dirichlet series as Mellin transforms

Summary

A Dirichlet series $F(s) = \sum_{n \ge 1} a(n) n^{-s}$ can be written as a Mellin transform in two standard ways. First, $\Gamma(s) F(s)$ is the Mellin transform of the exponential sum $\sum_n a(n) e^{-nx}$. Second, $F(s)/s$ is the Mellin transform at $-s$ of the summatory function $A(x) = \sum_{n \le x} a(n)$. These identities let one study Dirichlet series with the tools of integral transforms: analytic continuation, inversion formulas such as Perron's formula, and asymptotics through residues.