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Minimal zero-free regions for results on primes between consecutive perfect $k$th powers

Ethan Simpson Lee

Abstract

We compute minimal zero-free regions for the Riemann zeta-function of the Littlewood form which ensure there is always a prime between consecutive perfect $k$th powers. Our computations cover powers $k\geq 70$ and quantify how far we are away from proving certain milestones toward an infamous open problem (Legendre's conjecture).

Minimal zero-free regions for results on primes between consecutive perfect $k$th powers

Abstract

We compute minimal zero-free regions for the Riemann zeta-function of the Littlewood form which ensure there is always a prime between consecutive perfect th powers. Our computations cover powers and quantify how far we are away from proving certain milestones toward an infamous open problem (Legendre's conjecture).
Paper Structure (7 sections, 6 theorems, 41 equations, 5 tables)

This paper contains 7 sections, 6 theorems, 41 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Riemann zeta-function
  3. Zero-free regions
  4. Zero-density estimates
  5. Primes Between Consecutive Powers
  6. Parametrised bounds
  7. Proof of main results

Key Result

Theorem 1.1

If $n$ is sufficiently large, then there is at least one prime in the interval $(n^{3}, (n+1)^{3})$.

Theorems & Definitions (13)

  • Theorem 1.1: Ingham
  • Theorem 1.2
  • Remark
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark
  • Proposition 3.3
  • proof
  • ...and 3 more