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Density results for $r$-gaps between zeros of the Riemann zeta-function

Tianyu Zhao

Abstract

Let $0<γ_1\leq γ_2\leq \ldots$ denote the positive ordinates of the non-trivial zeros of the Riemann zeta-function. A result first announced by Selberg states that there exist absolute constants $Θ, \vartheta>0$ such that for each $r\in \mathbb{N}$, \[ \limsup_{n\to \infty}\frac{γ_{n+r}-γ_n}{2πr/\log γ_n}\geq 1+\fracΘ{r^α} \qquad \text{and}\qquad \liminf_{n\to \infty}\frac{γ_{n+r}-γ_n}{2πr/\log γ_n}\leq 1-\frac{\vartheta}{r^α} \] where $α$ may be taken as $2/3$, or as $1/2$ if one assumes the Riemann hypothesis. This was recently proved by Conrey and Turnage-Butterbaugh under RH and by Inoue unconditionally. We prove that in fact a positive proportion of $r$-gaps are large (and small) to the above extent, and we provide explicit estimates for the sizes and proportions of these gaps. In the case $r=1$, this quantitatively improves an unconditional result of Simonič, Trudgian and Turnage-Butterbaugh.

Density results for $r$-gaps between zeros of the Riemann zeta-function

Abstract

Let denote the positive ordinates of the non-trivial zeros of the Riemann zeta-function. A result first announced by Selberg states that there exist absolute constants such that for each , where may be taken as , or as if one assumes the Riemann hypothesis. This was recently proved by Conrey and Turnage-Butterbaugh under RH and by Inoue unconditionally. We prove that in fact a positive proportion of -gaps are large (and small) to the above extent, and we provide explicit estimates for the sizes and proportions of these gaps. In the case , this quantitatively improves an unconditional result of Simonič, Trudgian and Turnage-Butterbaugh.
Paper Structure (13 sections, 13 theorems, 148 equations)

This paper contains 13 sections, 13 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Reduction to Theorem \ref{['theorem measure S(t+h)-S(t)']} and setup for the proof
  3. From $S(t+h)-S(t)$ to zero gaps
  4. Setting up the proof of Theorem \ref{['theorem measure S(t+h)-S(t)']}
  5. Back to $r$-gaps
  6. Moments of $W(t)$
  7. Moments of $S(t+h)-S(t)$
  8. Moments of $R(t)$
  9. Completing the Proofs of Theorems \ref{['theorem measure S(t+h)-S(t)']} and \ref{['theorem 2']}
  10. Proof of Theorem \ref{['theorem measure S(t+h)-S(t)']}
  11. Proof of Theorem \ref{['theorem 2']}
  12. The unconditional case
  13. The conditional case

Key Result

Theorem 1

There exist absolute constants $\Theta,\vartheta>0$ such that for all $r\geq 1$, where one may take $\alpha=2/3$ unconditionally and $\alpha=1/2$ assuming RH.

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 14 more