Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Small gaps between primes or almost primes

Authors: D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim
Journal: Trans. Amer. Math. Soc. 361 (2009), 5285-5330
MSC (2000): Primary 11N25; Secondary 11N36
Published electronically: May 27, 2009
MathSciNet review: 2515812
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Abstract: Let $ p_n$ denote the $ n^{{\rm th}}$ prime. Goldston, Pintz, and Yıldırım recently proved that

$\displaystyle \liminf_{n\to \infty} \frac{(p_{n+1}-p_n)}{\log p_n} =0. $

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^{{\rm th}}$ number that is a product of exactly two distinct primes. We prove that

$\displaystyle \liminf_{n\to \infty} (q_{n+1}-q_n) \le 26. $

If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to $ 6$.

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Additional Information

D. A. Goldston
Affiliation: Department of Mathematics, San Jose State University, San Jose, California 95192

S. W. Graham
Affiliation: Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48859

J. Pintz
Affiliation: Rényi Mathematical Institute of the Hungarian Academy of Sciences, H-1053 Budapest, Realtanoda u. 13–15, Hungary

C. Y. Yildirim
Affiliation: Department of Mathematics, Bogaziçi University, Istanbul 34342, Turkey – and – Feza Gürsey Enstitüsü, Çengelköy, Istanbul, P.K. 6, 81220, Turkey

Keywords: Primes, almost primes, gaps, Selberg's sieve, applications of sieve methods
Received by editor(s): September 17, 2007
Published electronically: May 27, 2009
Additional Notes: The first author was supported by NSF grant DMS-0300563, the NSF Focused Research Group grant 0244660, and the American Institute of Mathematics.
The second author was supported by a sabbatical leave from Central Michigan University and by NSF grant DMS-070193.
The third author was supported by OTKA grants No. 43623, 49693, 67676 and the Balaton program.
The fourth author was supported by TÜBİTAK
Article copyright: © Copyright 2009 American Mathematical Society