On $r$-gaps between zeros of the Riemann zeta-function

TL;DR

Under the Riemann Hypothesis, the paper proves the existence of infinitely many large gaps between zeros of the Riemann zeta-function, demonstrating fluctuations around the average spacing with explicit bounds.

Contribution

It provides a rigorous proof of large and small gaps between zeros of the zeta-function, refining previous unproven claims and proposing a framework for further analysis.

Findings

Existence of infinitely many large gaps between zeros

Existence of infinitely many small gaps between zeros

Explicit bounds involving nd re established

Abstract

Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many large natural numbers $n$ such that $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) > r + \Theta\sqrt{r}$ and $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) < r - \vartheta\sqrt{r}$ for explicit absolute positive constants $\Theta$ and $\vartheta$, where $\gamma$ denotes an ordinate of a zero of the Riemann zeta-function on the critical line. Selberg published announcements of this result several times but did not include a proof. We also suggest a general framework which might lead to stronger statements concerning the vertical distribution of nontrivial zeros of the Riemann zeta-function.