Differential transcendence of a class of generalized Dirichlet series

Abstract

We investigate differential transcendence properties for a generalized Dirichlet series of the form $\sum_{n=0}^\infty a_n\lambda_n^{-s}$. Our treatment of this series is purely algebraic and does not rely on any analytic properties of generalized Dirichlet series. We establish differential transcendence theorems for a certain class of generalized Dirichlet series. These results imply that the Hurwits zeta-function $\zeta(s,a)$ does not satisfy an algebraic differential equation with complex coefficients.