Dirichlet's series associated with some power series

Abstract

We study the singularities of the Dirichlet series$D\left(s\right)=\sum _{n\ge 1}\frac{h_n}{n^s}$ when the power series $H\left(z\right)=\sum _{n\ge 0}{h}_n{z}^n$ has a singularity of general type at the point $z_0=1$. This extends a recent result of L.M. Navas, J. Ruiz, and J.L. Varona, and connects to foundational ideas explored by Hardy, Fekete, and others. The tools employed include classical methods from the theory of Dirichlet series, particularly those often used in connection with the Riemann zeta function, namely the Mellin transform and splitting methods. These techniques were also used by Navas, Ruiz, and Varona. The polylogarithm function plays a fundamental role in this work.