Acta Mathematica

Primes in tuples II

Daniel A. Goldston, János Pintz, and Cem Yalçin Yıldırım

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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 pn denotes the nth prime. Since on average pn+1pn 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 pp′ between primes which includes the small gap result above.


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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Acta Math., Volume 204, Number 1 (2010), 1-47.

Received: 15 November 2007
Revised: 21 October 2008
First available in Project Euclid: 31 January 2017

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2010 © Institut Mittag-Leffler


Goldston, Daniel A.; Pintz, János; Yalçin Yıldırım, Cem. Primes in tuples II. Acta Math. 204 (2010), no. 1, 1--47. doi:10.1007/s11511-010-0044-9.

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