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Primes in Tuples I

D. A. Goldston, J. Pintz, C. Y. Yildirim

Abstract

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, \[ \liminf_{n\to \infty} \frac{p_{n+1}-p_n}{\log p_n} =0 .\] This last result will be considerably improved in a later paper.

Primes in Tuples I

Abstract

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, This last result will be considerably improved in a later paper.

Paper Structure

This paper contains 10 sections, 8 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. Approximating prime tuples
  3. Proofs of Theorems 1 and 2
  4. Further Remarks on Section 3
  5. Two Lemmas
  6. A special case of Proposition 1
  7. First Part of the Proof of Proposition 1
  8. Completion of the proof of Proposition 1: Evaluating an integral
  9. Proof of Proposition 2
  10. Proof of Theorem 3

Key Result

Theorem 1

Suppose the primes have level of distribution $\vartheta > 1/2$. Then there exists an explicitly calculable constant $C(\vartheta)$ depending only on $\vartheta$ such that any admissible $k$-tuple with $k \geq C(\vartheta)$ contains at least two primes infinitely often. Specifically, if $\vartheta \

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Proposition 2
  • Remark
  • Lemma 1
  • Lemma 2
  • Remark
  • Lemma 3