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

Abstract

For any m ≥ 1, let H m denote the quantity lim inf n→∞ (p n+m − p n ). A celebrated recent result of Zhang showed the finiteness of H 1 , with the explicit bound H 1 ≤ 70, 000, 000. This was then improved by us (the Polymath8 project) to H 1 ≤ 4680, and then by Maynard to H 1 ≤ 600, who also established for the first time a finiteness result for H m for m ≥ 2, and specifically that H m m 3 e 4m . If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H 1 ≤ 12, improving upon the previous bound H 1 ≤ 16 of Goldston, Pintz, and Yıldırım, as well as the bound H m 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 ≤ 246 unconditionally and H 1 ≤ 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, and also obtain a related disjunction asserting that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both. We also modify the 'parity problem' argument of Selberg to show that the H 1 ≤ 6 bound 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

Analytic Number Theory

“…, n + h k are prime. Obviously, k m ≥ m, and in [6] it was shown that one can take k 2 = 50 and k m ≪ e 3.82m .…”

Bombieri–vinogradov Theorems for Modular Forms and Applications

“…Then, by deriving lower bounds for defined in Theroem , Maynard established his groundbreaking work on small gaps between primes. Shortly after Maynard's work, the Polymath project applied a more dedicate numerical method to obtain better lower bounds for . Nowadays, one has …”

“…Now let us borrow a lower bound for from [, Theorem 23], i.e. for all , for some absolute c .…”

Some Comments on Multiple Discovery in Mathematics

“…The Polymath 8 project, for example, resulted in the publication [43], under the name D.H.J. Polymath, of significant improvements to bounds arising from Zhang and Maynard's small prime gaps breakthrough, improvements which could hardly have been achieved by one person working on their own, let alone two such simultaneously.…”