Zeros of Hurwitz zeta functions

Abstract

AU complex zeros of each Hurwitz zeta function are shown to lie in a vertical strip. Trivial real zeros analogous to those for the Riemann zeta function are found. Zeros of two particular Hurwitz zeta functions are calculated.

Interpolation and Sampling with Exponential Splines of Real Order

“…has no zeros for all Re q ∈ (0, 1) and a fixed a > 0. We require the following lemma which is based on a result in [15] for the case of real q.…”

“…Note that σ ≥ 1 + 1 + a 2 4π 2 > 2 and that Arg(ζ * ) ∈ (−π, π] (taking the negative real axis as a branch cut) and therefore, we need to impose condition (15) to ensure that Z(σ, q * ) 0 as ζ(σ, q * ) 0. Remark 6.…”

Complex B-splines and Hurwitz zeta functions

“…The aim of this article is to characterize nonempty open subsets of the complex plane for which (7) has no zeros for all α ∈ ] 0, 1 [. For these parameters s the interpolation property (5) is satisfied.…”

“…To find nonempty open subsets of the complex plane for which (7) has no zeros, we consider a more general sum of Hurwitz zeta functions. For s = σ + it and a real parameter α ∈ (0, 1) define For σ > 1, the Hurwitz zeta function to the parameter α ∈ (0, 1] can be extended by analytic continuation to all of C except for a simple pole at s = 1.…”

Moments of the Hurwitz zeta function on the critical line

“…However, for 0 < α < 1, α = 1/2, ζ(s, α) does not have an Euler product, and, in fact, the behaviour of its zeroes is very different from that of ζ(s). Spira [Spi76] showed that like ζ(s), ζ(s, α) may have trivial zeros on the negative real line, and also that ζ(s, α) is zero-free in the region ℜs ≥ 1 + α. It was classically known, due to Davenport and Heilbronn [DH36] for the cases of rational or transcendental α, and due to Cassels [Cas61] for the case of algebraic irrational α that if α = 1/2, 1, then ζ(s, α) always has a zero in the strip 1 < ℜs < 1 + δ for every δ > 0.…”