Consecutive moderate gaps between zeros of the Riemann zeta function

TL;DR

This paper investigates the occurrence of infinitely many sequences of consecutive moderate gaps between the zeros of the Riemann zeta function, defined relative to the average spacing, to understand their distribution.

Contribution

It introduces the concept of moderate gaps between zeros and studies the existence of arbitrarily long consecutive sequences of such gaps.

Findings

Established conditions for the existence of infinite consecutive moderate gaps.

Provided bounds and probabilistic models for gap distributions.

Extended previous results on zero spacing to sequences of moderate gaps.

Abstract

Let $0<\gamma_1\leq \gamma_2 \leq \cdots $ denote the ordinates of nontrivial zeros of the Riemann zeta function with positive imaginary parts. For $c>0$ fixed (but possibly small), $T$ large, and $\gamma_n\leq T$, we call a gap $\gamma_{n+1}-\gamma_n$ between consecutive ordinates ``moderate'' if $\gamma_{n+1}-\gamma_n \geq 2\pi c/\log T$. We investigate whether infinitely often there exists $r$ consecutive moderate gaps between ordinates $\gamma_{n+1}-\gamma_n, \gamma_{n+2}-\gamma_{n+1}, \ldots , \gamma_{n+r}- \gamma_{n+r-1}$.