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On limit points of the sequence of normalized prime gaps

William D. Banks, Tristan Freiberg, James Maynard

Abstract

Let $p_n$ denote the $n$th smallest prime number, and let $\boldsymbol{L}$ denote the set of limit points of the sequence $\{(p_{n+1} - p_n)/\log p_n\}_{n = 1}^{\infty}$ of normalized differences between consecutive primes. We show that for $k = 9$ and for any sequence of $k$ nonnegative real numbers $β_1 \le β_2 \le ... \le β_k$, at least one of the numbers $β_j - β_i$ ($1 \le i < j \le k$) belongs to $\boldsymbol{L}$. It follows at least $12.5%$ of all nonnegative real numbers belong to $\boldsymbol{L}$.

On limit points of the sequence of normalized prime gaps

Abstract

Let denote the th smallest prime number, and let denote the set of limit points of the sequence of normalized differences between consecutive primes. We show that for and for any sequence of nonnegative real numbers , at least one of the numbers () belongs to . It follows at least of all nonnegative real numbers belong to .

Paper Structure

This paper contains 5 sections, 5 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Outline of the proof
  4. A uniform Maynard--Tao theorem
  5. Preliminaries

Key Result

Theorem 1.1

Let $d_n = p_{n+1} - p_n$, where $p_n$ denotes the $n$th smallest prime, and let $\boldsymbol{L}$ be the set of limit points of $\left\{d_n/\log p_n\right\}_{n = 1}^{\infty}.$ For any sequence of $k = 9$ nonnegative real numbers $\beta_1 \leqslant \beta_2 \leqslant \cdots \leqslant \beta_k$, we have

Theorems & Definitions (5)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 4.1
  • Theorem 4.2: Modified Bombieri--Vinogradov theorem