Originally asked on MSE here: https://math.stackexchange.com/q/1780149/52694

Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the original function at every point.

Is there a corresponding notion for Dirichlet series? When is a function able to be decomposed into a Dirichlet series, and does a decomposition exist for any common functions?

To clarify, by Dirichlet series I mean a decomposition of a function $f(s)$ into the form

$f(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$

for complex $s$ and some complex sequence of $a_n$.