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Zero-free regions for the Riemann zeta function

Kevin Ford

TL;DR

This work delivers explicit improvements to Korobov–Vinogradov type zero-free regions for the Riemann zeta function by marrying an explicit near-1 growth bound for ζ(σ+it) with a Jensen-type zero-count mechanism. A novel two-vertical-lines zero detector, a tightening trigonometric mollifier inequality, and a carefully chosen compactly supported mollifier allow the author to obtain fully explicit VK-type zero-free regions for all |t|≥3 and, separately, sharper regions for large t via a VK framework. The results carry explicit constants and are designed to feed directly into explicit bounds for prime-counting and related arithmetic functions. A second result leverages Van der Corput bounds to push the explicit region further, demonstrating the method’s adaptability to different analytic inputs.

Abstract

We improve existing explicit bounds of Vinogradov-Korobov type for zero-free regions of the Riemann zeta function, both for large height t and for every t. A primary input is an explicit bound of the author (Proc. London Math. Soc. 85 (2002), 565-633) for the growth of $ζ(σ+it)$ for $σ$ near 1. Another ingredient is a kind of Jensen formula (Lemma 2.2) relating the growth of a function on vertical lines to a weighted count of its zeros inside a vertical strip. The latter was used recently in the weak subconvexity work of Soundararajan and Thorner (arXiv:1804.03654, Duke Math. J. 168, no. 7 (2019), 1231-1268).

Zero-free regions for the Riemann zeta function

TL;DR

This work delivers explicit improvements to Korobov–Vinogradov type zero-free regions for the Riemann zeta function by marrying an explicit near-1 growth bound for ζ(σ+it) with a Jensen-type zero-count mechanism. A novel two-vertical-lines zero detector, a tightening trigonometric mollifier inequality, and a carefully chosen compactly supported mollifier allow the author to obtain fully explicit VK-type zero-free regions for all |t|≥3 and, separately, sharper regions for large t via a VK framework. The results carry explicit constants and are designed to feed directly into explicit bounds for prime-counting and related arithmetic functions. A second result leverages Van der Corput bounds to push the explicit region further, demonstrating the method’s adaptability to different analytic inputs.

Abstract

We improve existing explicit bounds of Vinogradov-Korobov type for zero-free regions of the Riemann zeta function, both for large height t and for every t. A primary input is an explicit bound of the author (Proc. London Math. Soc. 85 (2002), 565-633) for the growth of for near 1. Another ingredient is a kind of Jensen formula (Lemma 2.2) relating the growth of a function on vertical lines to a weighted count of its zeros inside a vertical strip. The latter was used recently in the weak subconvexity work of Soundararajan and Thorner (arXiv:1804.03654, Duke Math. J. 168, no. 7 (2019), 1231-1268).

Paper Structure

This paper contains 9 sections, 21 theorems, 171 equations, 1 table.

Table of Contents

  1. Introduction
  2. The zero detector
  3. Bounds for $\zeta(s)$
  4. Detecting zeros of $\zeta(s)$
  5. A trigonometric inequality
  6. The functions $f$, $F$ and $K$
  7. An inequality for the real part of a zero
  8. The proof of Theorem 2
  9. The proof of Theorem 3

Key Result

Theorem 1

If eq:Richert holds with a certain constant $B$, then for large $|t|$, $\zeta(\beta+it) \ne 0$ for

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • ...and 26 more