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Zero-free regions inspired by work of Heath-Brown

Chiara Bellotti, Tim Trudgian, Andrew Yang

Abstract

We prove a new explicit zero-free region for the Riemann zeta-function, drawing substantially on Heath-Brown's seminal work on Linnik's constant. Using these ideas we are able to prove that $ζ(σ+ it)\ne 0$ whenever $t\geq 3$ and $σ\geq 1- 1/(4.896\log t)$.

Zero-free regions inspired by work of Heath-Brown

Abstract

We prove a new explicit zero-free region for the Riemann zeta-function, drawing substantially on Heath-Brown's seminal work on Linnik's constant. Using these ideas we are able to prove that whenever and .
Paper Structure (5 sections, 17 theorems, 154 equations, 1 table)

This paper contains 5 sections, 17 theorems, 154 equations, 1 table.

Table of Contents

  1. Introduction
  2. Reasons for choosing $f$
  3. The test function
  4. The trigonometric polynomial
  5. Proof of Lemma \ref{['classical_main_lem']}

Key Result

Theorem 1

If $t \ge 3$ then $\zeta(\sigma + it) \ne 0$ in the region $\sigma > 1 - 1/(4.896 \log t).$

Theorems & Definitions (30)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Lemma 2
  • Definition 1: The functions $w$ and $W$
  • Definition 2: The functions $f$ and $F$
  • Lemma 3
  • proof
  • Lemma 4: Ford ford_zero_2002 § 7
  • Lemma 5: Non-negativity property
  • ...and 20 more