A note on the zeros of generalized Hurwitz zeta functions

Abstract

Given a function f (n) periodic of period q ≥ 1 and an irrational number 0 < α ≤ 1, Chatterjee and Gun (cf. [4]) proved that the series F (s, f, α) = ∞ n=0 f (n) (n+α) s has infinitely many zeros for σ > 1 when α is transcendental and F (s, f, α) has a pole at s = 1, or when α is algebraic irrational and c = max f (n) min f (n) < 1.15. In this note, we prove that the result holds in full generality.

On the linear twist of degree $1$ functions in the extended Selberg class

“…If α is irrational, F * (s, 0, ±αq) can be seen as generalized Hurwitz zeta functions with periodic coefficients. Thus, by our result on the zeros of generalized Hurwitz zeta functions [17], we deduce that they have infinitely many zeros for σ > 1. Therefore, if σ < 0, F * (1 − s, 0, ±αq) have infinitely many zeros (cf.…”