arXiv 1407.48972014-07math.NT

Selberg筛法的变体和包含许多素数的有界区间

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

D. H. J. Polymath

备注：: 79 pages, 1 figure. A minor issue with the proof of (117) has been fixed
发表信息：: Research in the Mathematical Sciences 1:12 (2014)

摘要：

对于任何$m \geq 1$，令$H_m$表示数量$\liminf_{n \to \infty} (p_{n+m}-p_n)$。Zhang最近的一个著名结果证明了$H_1$的有限性，并给出了$H_1 \leq 70000000$的显式界.然后我们（Polymath 8项目）将其改进为$H_1 \leq 4680$，然后由Maynard改进为$H_1 \leq 600$，他还首次为$m \geq 2$建立了$H_m$的有限性结果，特别是$H_m \ll m^3 e^{4m}$。如果我们也假设艾略特-哈伯鲁猜想，梅纳德得到了界$H_1 \leq 12$，改进了Goldston、Pintz和Yıldırım之前的界$H_1 \leq 16$，以及界$H_m \ll m^3 e^{2m}$。在本文中，我们通过进一步推广Selberg筛，并通过进行更广泛的数值计算，扩展了Maynard方法。因此，我们可以无条件地得到有界的$H_1 \leq 246$，以及在广义Elliott-Halbertmann猜想的假设下得到有界的$H_1 \leq 6$。事实上，根据后者的猜想，我们表明更强的声明，任何容许三重$(h_1,h_2,h_3)$，有无穷多个$n$，其中至少有两个$n+h_1,n+h_2,n+h_3$是素数。我们修改的“奇偶校验问题”的论点塞尔伯格表明，这一结果是最好的可能，人们可以从纯粹的筛理论的考虑。对于更大的$m$，我们使用我们的项目以前获得的分布结果，以获得无条件渐近界$H_m \ll m e^{(4-\frac{24}{181})m}$，或$H_m \ll m e^{2m}$的假设下的Elliott-Halbernut猜想。当$m=2,3,4,5$时，我们也得到了$H_m$的明确的上界。

英文摘要：

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 Yıldırım, 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 $m=2,3,4,5$.

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