Unconditional and Conditional Large Gaps between the zeros of the Riemann Zeta-Function

TL;DR

This paper derives new lower bounds for the gaps between zeros of the Riemann zeta function using inequalities and moment predictions, improving previous bounds and providing both unconditional and conditional results.

Contribution

It introduces new unconditional lower bounds using Opial inequalities and conditional bounds based on moment conjectures, advancing understanding of zero spacing.

Findings

Unconditional lower bounds for zero gaps established.

Conditional bounds suggest zeros often differ by at least 6.1392 times the average.

Improves previous lower bound estimate from 4.7147 to 6.1392.

Abstract

In this paper, first by employing inequalities derived from the Opial inequality due to David Boyd with best constant, we will establish new unconditional lower bounds for the gaps between the zeros of the Riemann zeta function. Second, on the hypothesis that the moments of the Hardy Z-function and its derivatives are correctly predicted, we establish some explicit formulae for the lower bounds of the gaps between the zeros and use them to establish some new conditional bounds. In particular it is proved that the consecutive nontrivial zeros often differ by at least 6.1392 (conditionally) times the average spacing. This value improves the value 4.71474396 that has been derived in the literature.