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Dense clusters of primes in subsets

James Maynard

Abstract

We prove a generalization of the author's work to show that any subset of the primes which is `well-distributed' in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the parameters. As applications, we show there are infinitely many intervals of length $(\log{x})^ε$ containing $\gg_ε\log\log{x}$ primes, and show lower bounds of the correct order of magnitude for the number of strings of $m$ congruent primes with $p_{n+m}-p_n\le ε\log{x}$.

Dense clusters of primes in subsets

Abstract

We prove a generalization of the author's work to show that any subset of the primes which is `well-distributed' in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the parameters. As applications, we show there are infinitely many intervals of length containing primes, and show lower bounds of the correct order of magnitude for the number of strings of congruent primes with .

Paper Structure

This paper contains 10 sections, 17 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Well-distributed sets
  3. Main Results
  4. Notation
  5. Outline
  6. Proof of theorems \ref{['thrm:MainTheorem']}, \ref{['thrm:UniformSmallGaps']}, \ref{['thrm:AveragedShiu']}, \ref{['thrm:ShortIntervals']} and \ref{['thrm:Chebotarev']}
  7. Initial Considerations
  8. Preparatory Lemmas
  9. Proof of Propositions
  10. Acknowledgements

Key Result

Theorem 3.1

Let $\alpha>0$ and $0<\theta<1$. Let $\mathcal{A}$ be a set of integers, $\mathcal{P}$ a set of primes, $\mathcal{L}=\{L_1,\dots,L_k\}$ an admissible set of $k$ linear functions, and $B,x$ integers. Let the coefficients $L_i(n)=a_in+b_i\in\mathcal{L}$ satisfy $1\le a_i,b_i\le x^{\alpha}$ for all $1\ then Moreover, if $\mathcal{P}=\mathbb{P}$, $k\le (\log{x})^{1/5}$ and all $L\in\mathcal{L}$ have

Theorems & Definitions (18)

  • Conjecture : Prime $k$-tuples conjecture
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Proposition 6.1
  • Lemma 8.1
  • Lemma 8.2
  • Lemma 8.3
  • ...and 8 more