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Primes in Tuples II

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

Abstract

We prove that there are infinitely often pairs of primes much closer than the average spacing between primes - almost within the square root of the average spacing. We actually prove a more general result concerning the set of values taken on by the differences between primes.

Primes in Tuples II

Abstract

We prove that there are infinitely often pairs of primes much closer than the average spacing between primes - almost within the square root of the average spacing. We actually prove a more general result concerning the set of values taken on by the differences between primes.

Paper Structure

This paper contains 15 sections, 22 theorems, 261 equations.

Table of Contents

  1. Introduction
  2. A generalization of Theorem 1
  3. Some Initial Considerations
  4. Two basic theorems
  5. Lemmas
  6. Preparation for the Proof of Theorem \ref{['ujth:1']}
  7. Estimates of the partial derivatives of $D(s_1,s_2)$
  8. Contribution of the residue at $s_1 = s_2 = 0$
  9. Estimate of the integral $I_2$
  10. Comparison of $D(s, -s)$ and $D(0,0)$
  11. Estimate of $I_2$. Evaluation of $I$
  12. A Bombieri--Vinogradov type theorem
  13. Proof of Theorem \ref{['ujth:2']}
  14. The sum of the singular series $\mathfrak S(\mathcal{H})$
  15. Proof of Theorem \ref{['th:A']}

Key Result

Theorem 1

The differences of consecutive primes satisfy

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 16 more