On the zeros of generalized Hurwitz zeta functions

Abstract

Abstract. In this note, we prove the existence of infinitely many zeros of certain generalized Hurwitz zeta functions in the domain of absolute convergence. This is a generalization of a classical problem of Davenport, Heilbronn and Cassels about the zeros of the Hurwitz zeta function.

A note on the zeros of generalized Hurwitz zeta functions

“…Observe that such a σ exists by (4). Now, for p | a or p | (n + α)a with n ≤ N 1 we choose ϕ(p) = 1.…”

“…As for ζ(s, α), F (s, f, α) is absolutely convergent for σ > 1 and it admits a meromorphic continuation to the whole complex plane (see e.g. [4]).…”

“…In [4], Chatterjee and Gun assume that f (n) is positive valued and prove that F (s, f, α) has infinitely many zeros in the half-plane σ > 1 if α is transcendental and F (s, f, α) has a pole at s = 1, or if α is algebraic irrational and…”

“…Then, the assumption on the existence of the pole can be avoided and one can proceed as in [5] or [4]. Thus, we focus on the case of α algebraic irrational.…”

“…

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.

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