Small gaps between primes or almost primes

Goldston, D.; Graham, S.; Pintz, J.; Yıldırım, C.

doi:10.1090/S0002-9947-09-04788-6

Small gaps between primes or almost primes
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by D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yıldırım PDF

Trans. Amer. Math. Soc. 361 (2009), 5285-5330 Request permission

Abstract:

Let $p_n$ denote the $n^{\textrm {th}}$ prime. Goldston, Pintz, and Yıldırım recently proved that \begin{equation*} \liminf _{n\to \infty } \frac {(p_{n+1}-p_n)}{\log p_n} =0. \end{equation*} We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let $q_n$ denote the $n^{\textrm {th}}$ number that is a product of exactly two distinct primes. We prove that \begin{equation*} \liminf _{n\to \infty } (q_{n+1}-q_n) \le 26. \end{equation*} If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to $6$.

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