Which complex maps with branch cuts have a representation by Dirichlet series?
I am aware of the work of A.F. Leont'ev on general Dirichlet series, and the theorems of representation of analytic functions. A related question with a good answer can be find here, for example. I can also easily obtain representation by Dirichlet series of certain meromorphic maps using the Mellin transformation [Flajolet,Gourdon,Dumas, Mellin transforms and asymptotics: harmonic sums. Theoret. Comput. Sci. 144 (1995)]. On the other hand, I have not found any reference for the case of complex maps with branch cuts. I am aware, however, that using methods in [A. Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc. B. 18 (1954)], it is possible to go in the opposite direction of the question, i.e., building Dirichlet series with analytic continuation that are complex maps with branch cuts. In any case, I only know how to do this for very special Dirichlet series.
