Differential transcendence of a class of generalized Dirichlet series

We investigate differential transcendence properties for a generalized Dirichlet series of the form ∑^∞_n=0 a_nλ^-s_n. 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 ζ(s, a) does not satisfy an algebraic differential equation with complex coefficients.