Analytic continuation of the series $\sum (m+nz)^{-s}$

Abstract:

The series $\Sigma {(m + nz)^{ - s}},m,n$ ranging over all integers except both zero, for $s$ an integer greater than two is well known from the theory of elliptic functions and modular forms. In this paper, we show that this series defines an analytic function $(z,s)$ for $\operatorname {Im} z > 0$ and $\operatorname {Re} s > 2$ which has an analytic continuation to all values of $s$. It is then shown that $G$ satisfies a functional equation under the transformation $z \to - 1/z$, and finally as an application some numerical results are obtained.