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

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.

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