A small improvement in the gaps between consecutive zeros of the Riemann zeta-function

Abstract

Feng and Wu introduced a new general coefficient sequence into Montgomery and Odlyzko's method for exhibiting irregularity in the gaps between consecutive zeros of ζ (s) assuming the Riemann hypothesis. They used a special case of their sequence to improve upon earlier results on the gaps. In this paper we consider a general sequence related to that of Feng and Wu, and introduce a somewhat less general sequence {a n } for which we write the Montgomery-Odlyzko expressions explicitly. As an application, we give the following slight improvement of Feng and Wu's result: infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at most 0.515396 times the average spacing and infinitely often they differ by at least 2.7328 times the average spacing.

Gaps between zeros of the Riemann zeta-function

“…A different method of Mueller [25] has been used in a number of papers to prove lower bounds for λ conditional upon RH and its generalizations, see [7,11,13,14,16,24,26,27]. Our result in Theorem 1.1 that λ > 3.18 assuming RH supersedes all of these previous results.…”

On r‐gaps between zeros of the Riemann zeta‐function

“…With δ sufficiently small, T and r sufficiently large, these choices guarantee that h + (c r ) < r. We now consider small gaps for r sufficiently large. We begin with (10) As before, the choice B = 1.502243 yields ϑ = 0.9065. With δ sufficiently small, T and r sufficiently large, these choices guarantee that h − (c r ) > r.…”

A limitation on proving the existence of small gaps between zeta-zeros

“…We refer the reader to [4,8] for the history of this problem. The best current results under RH are: µ ≤ 0.515396 by Preobrazhenski ȋ [7] and λ ≥ 3.18 by Bui and Milinovich [1]. The result of [7] is based on a method introduced by Montgomery and Odlyzko [6].…”

“…The best current results under RH are: µ ≤ 0.515396 by Preobrazhenski ȋ [7] and λ ≥ 3.18 by Bui and Milinovich [1]. The result of [7] is based on a method introduced by Montgomery and Odlyzko [6]. Define, for T ≥ 2, c > 0, and a k a sequence of complex numbers, (1) and y = T 1−δ for some small δ > 0.…”