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Small gaps between primes

James Maynard

Abstract

We introduce a refinement of the GPY sieve method for studying prime $k$-tuples and small gaps between primes. This refinement avoids previous limitations of the method, and allows us to show that for each $k$, the prime $k$-tuples conjecture holds for a positive proportion of admissible $k$-tuples. In particular, $\liminf_{n}(p_{n+m}-p_n)<\infty$ for any integer $m$. We also show that $\liminf(p_{n+1}-p_n)\le 600$, and, if we assume the Elliott-Halberstam conjecture, that $\liminf_n(p_{n+1}-p_n)\le 12$ and $\liminf_n (p_{n+2}-p_n)\le 600$.

Small gaps between primes

Abstract

We introduce a refinement of the GPY sieve method for studying prime -tuples and small gaps between primes. This refinement avoids previous limitations of the method, and allows us to show that for each , the prime -tuples conjecture holds for a positive proportion of admissible -tuples. In particular, for any integer . We also show that , and, if we assume the Elliott-Halberstam conjecture, that and .

Paper Structure

This paper contains 9 sections, 16 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. An improved GPY sieve method
  3. Notation
  4. Outline of the proof
  5. Selberg sieve manipulations
  6. Smooth choice of y
  7. Choice of smooth weight for large k
  8. Choice of weight for small k
  9. Acknowledgements

Key Result

Theorem 1.1

Let $m\in\mathbb{N}$. We have

Theorems & Definitions (22)

  • Conjecture : Prime $k$-tuples conjecture
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Lemma 5.1
  • Lemma 5.2
  • ...and 12 more