Variants of the Selberg sieve, and bounded intervals containing many primes

TL;DR

This paper advances sieve methods to improve bounds on the minimal gaps between primes, achieving new unconditional and conditional results, and extends the understanding of prime distributions within bounded intervals.

Contribution

It generalizes the Selberg sieve and performs extensive numerical calculations to improve bounds on prime gaps, including new results under the generalized Elliott-Halberstam conjecture.

Findings

Unconditionally, H_1  246

Under the generalized Elliott-Halberstam conjecture, H_1  6

Established bounds for H_m for m=2,3,4,5

Abstract

For any $m \geq 1$, let $H_m$ denote the quantity $\liminf_{n \to \infty} (p_{n+m}-p_n)$. A celebrated recent result of Zhang showed the finiteness of $H_1$, with the explicit bound $H_1 \leq 70000000$. This was then improved by us (the Polymath8 project) to $H_1 \leq 4680$, and then by Maynard to $H_1 \leq 600$, who also established for the first time a finiteness result for $H_m$ for $m \geq 2$, and specifically that $H_m \ll m^3 e^{4m}$. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound $H_1 \leq 12$, improving upon the previous bound $H_1 \leq 16$ of Goldston, Pintz, and Y{\i}ld{\i}r{\i}m, as well as the bound $H_m \ll m^3 e^{2m}$. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further, and by performing more extensive numerical calculations. As a consequence, we can obtain the bound $H_1 \leq 246$ unconditionally, and $H_1 \leq 6$ under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture we show the stronger statement that for any admissible triple $(h_1,h_2,h_3)$, there are infinitely many $n$ for which at least two of $n+h_1,n+h_2,n+h_3$ are prime. We modify the "parity problem" argument of Selberg to show that this result is the best possible that one can obtain from purely sieve-theoretic considerations. For larger $m$, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound $H_m \ll m e^{(4-\frac{24}{181})m}$, or $H_m \ll m e^{2m}$ under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for $H_m$ when $m=2,3,4,5$.