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Large gaps between primes

James Maynard

Abstract

We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our proof works by incorporating recent progress in sieve methods related to small gaps between primes into the Erdos-Rankin construction. This answers a well-known question of Erdos.

Large gaps between primes

Abstract

We show that there exists pairs of consecutive primes less than whose difference is larger than for any fixed . Our proof works by incorporating recent progress in sieve methods related to small gaps between primes into the Erdos-Rankin construction. This answers a well-known question of Erdos.

Paper Structure

This paper contains 7 sections, 8 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. The Erdős-Rankin construction
  3. The probabilistic method
  4. GPY Probabilities
  5. Setup
  6. Sieve estimates
  7. Acknowledgments

Key Result

Theorem 1

We have

Theorems & Definitions (9)

  • Theorem 1
  • Remark
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Proposition 5
  • Lemma 6
  • Lemma 7
  • Lemma 8