Xiao-Feng Xie, Ph.D.

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xie@wiomax.com

Work with the Polymath project was published on Research in the Mathematical Sciences

The work with the Polymath project was published on Research in the Mathematical Sciences.

D. H. J. Polymath, et al. Variants of the Selberg sieve, and bounded intervals containing many primes.Research in the Mathematical Sciences, 2014, 1(12): 1-83. [DOI]

Here is the list of Polymath8b authors (arranged in alphabetical order of surname): Ignace Bogaert, Aubrey de Grey, Gergely Harcos, Emmanuel Kowalski, Philippe Michel, James Maynard, Paul Nelson, Pace Nielsen, Eytan Paldi, Andrew V. Sutherland, Terence Tao, Xiao-Feng Xie

Abstract: For any $m \geq 1$ , let $H_m$ denote the quantity $\liminf_{n \to \infty} (p_{n+m}-p_n)$ , where $p_n$ is the $n^{\text{th}}$ prime. 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, 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 $H_1 \leq 6$ 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 $H_m \ll m e^{(4-\frac{28}{157})m}$ , 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 $m=2,3,4,5$ .