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 $6$.References
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Additional Information
- D. A. Goldston
- Affiliation: Department of Mathematics, San Jose State University, San Jose, California 95192
- MR Author ID: 74830
- ORCID: 0000-0002-6319-2367
- Email: goldston@math.sjsu.edu
- S. W. Graham
- Affiliation: Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48859
- MR Author ID: 76030
- Email: graha1sw@cmich.edu
- J. Pintz
- Affiliation: Rényi Mathematical Institute of the Hungarian Academy of Sciences, H-1053 Budapest, Realtanoda u. 13–15, Hungary
- Email: pintz@renyi.hu
- C. Y. Yıldırım
- Affiliation: Department of Mathematics, Bog̃aziçi University, Istanbul 34342, Turkey – and – Feza Gürsey Enstitüsü, Çengelköy, Istanbul, P.K. 6, 81220, Turkey
- Email: yalciny@boun.edu.tr
- 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 - © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 5285-5330
- MSC (2000): Primary 11N25; Secondary 11N36
- DOI: https://doi.org/10.1090/S0002-9947-09-04788-6
- MathSciNet review: 2515812