Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Explicit extreme values of the argument of the Riemann zeta-function

Shōta Inoue, Hirotaka Kobayashi, Yuichiro Toma

TL;DR

This work studies explicit extreme values of the argument $S(t)$ of the Riemann zeta-function on the critical line, focusing on short-interval differences $S(t+h)-S(t)$ and their connection to gaps between zeta zeros. By deploying the resonance method with Dirichlet-polynomial resonators, the authors derive RH-conditional lower bounds for these short-interval extrema and translate them into explicit bounds for the normalized $r$-gaps $\lambda_r$ and $\mu_r$, improving constants toward the conjectured extremal values. They obtain, for large $r$, $\lambda_r \ge 1 + \sqrt{2}/\sqrt{r} - C_1 (\log r)^{3/2}/r$ and $\mu_r \le 1 - \sqrt{2}/\sqrt{r} + C_2 (\log r)^{3/2}/r$, under RH, thereby strengthening previous results. The paper also analyzes limitations of the Montgomery–Odlyzko approach, proving barriers such as $\lambda_1 \ge 3.022$ and $\mu_1 \le 0.508$, and discusses the optimality of the $\sqrt{2}$ constant within that framework.

Abstract

We investigate explicit extreme values of the argument of the Riemann zeta-function in short intervals. As an application, we improve the result of Conrey and Turnage-Butterbaugh concerning $r$-gaps between zeros of the Riemann zeta-function.

Explicit extreme values of the argument of the Riemann zeta-function

TL;DR

This work studies explicit extreme values of the argument of the Riemann zeta-function on the critical line, focusing on short-interval differences and their connection to gaps between zeta zeros. By deploying the resonance method with Dirichlet-polynomial resonators, the authors derive RH-conditional lower bounds for these short-interval extrema and translate them into explicit bounds for the normalized -gaps and , improving constants toward the conjectured extremal values. They obtain, for large , and , under RH, thereby strengthening previous results. The paper also analyzes limitations of the Montgomery–Odlyzko approach, proving barriers such as and , and discusses the optimality of the constant within that framework.

Abstract

We investigate explicit extreme values of the argument of the Riemann zeta-function in short intervals. As an application, we improve the result of Conrey and Turnage-Butterbaugh concerning -gaps between zeros of the Riemann zeta-function.
Paper Structure (9 sections, 11 theorems, 74 equations)

This paper contains 9 sections, 11 theorems, 74 equations.

Table of Contents

  1. Introduction and statement of results
  2. A relationship between $S(t)$ and large/small gaps of consecutive $r$ zeros
  3. Preliminaries
  4. Proof of \ref{['EFCVStRtPhi']}
  5. Proof of \ref{['LBStvR']}
  6. Proof of \ref{['ORSi']}
  7. Proof of \ref{['LGZ']}
  8. Proof of \ref{['LGZMO']}
  9. Limitations of $\lambda_{1}$ and $\mu_{1}$ deduced from \ref{['LGZMO']}

Key Result

Theorem 1

Assume RH. For any large $T$ and any $h \in [C / \log{T}, c / \log\log{T}]$ with positive constants $C$ large and $c$ small, we have where the error term $E$ satisfies

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • proof
  • Remark 1
  • Proposition 2
  • Proposition 3
  • Lemma 3.1
  • proof
  • ...and 8 more