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Primes in tuples II

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Acta Mathematica

Abstract

We prove that

$$ \mathop{ \lim \inf}\limits_{n \rightarrow \infty} \frac{p_{n+1}-p_{n}}{\sqrt{\log p_{n}} \left(\log \log p_{n}\right)^{2}}< \infty, $$

where p n denotes the nth prime. Since on average p n+1−p n is asymptotically log n , this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences p−p′ between primes which includes the small gap result above.

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Authors and Affiliations

  1. Department of Mathematics, San José State University, One Washington Square, San José, CA, 95192, U.S.A.

    Daniel A. Goldston

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, HU-1053, Budapest, Hungary

    János Pintz

  3. Department of Mathematics, Bog̃aziçi University, Bebek Istanbul, 34342, Turkey

    Cem Yalçin Yıldırım

Authors
  1. Daniel A. Goldston
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  2. János Pintz
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  3. Cem Yalçin Yıldırım
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Corresponding author

Correspondence to Daniel A. Goldston.

Additional information

The first author was supported in part by an NSF Grant, the second author by OTKA grants No. K 67676, T 43623, T 49693 and the Balaton program, the third author by TÜBITAK.

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Goldston, D.A., Pintz, J. & Yalçin Yıldırım, C. Primes in tuples II. Acta Math 204, 1–47 (2010). https://doi.org/10.1007/s11511-010-0044-9

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  • DOI: https://doi.org/10.1007/s11511-010-0044-9

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