Dirichlet series decomposition of arbitrary function

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 $a_n$.

• This seems to address your questions: mathoverflow.net/questions/30975/… – M.G. May 13 '16 at 18:59
• You have to define exactly what you mean by Dirichlet series: this word can have several different meanings. What are your exponents? Real? Complex? $\log n$? – Alexandre Eremenko May 14 '16 at 2:06
• Yeah, I meant a "classic" Dirichlet series, for lack of a better term. So $\log n$. – Mike Battaglia May 14 '16 at 4:22
• Functions that can be expanded into a classical Dirichlet series make a very special class, for example they are bounded on every vertical line where the series is absolutely convergent, they are almost periodic on these lines, with prescribed spectrum, etc. – Alexandre Eremenko May 14 '16 at 12:32