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

TL;DR

This paper refines the bounds on the gaps between consecutive zeros of the Riemann zeta-function, providing a slight improvement over previous results under the Riemann Hypothesis.

Contribution

It introduces a less general coefficient sequence and explicitly computes Montgomery-Odlyzko expressions to improve gap bounds.

Findings

Infinitely often, zeros differ by at most 0.515396 times the average spacing.

Provides explicit formulas for a new coefficient sequence.

Achieves a marginal improvement in zero gap bounds.

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 $\zeta(s)$ 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 an equivalent form of the general sequence 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.