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2010 Primes in tuples II
Daniel A. Goldston, János Pintz, Cem Yalçin Yıldırım
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Acta Math. 204(1): 1-47 (2010). DOI: 10.1007/s11511-010-0044-9


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.

Funding Statement

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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Daniel A. Goldston. János Pintz. Cem Yalçin Yıldırım. "Primes in tuples II." Acta Math. 204 (1) 1 - 47, 2010.


Received: 15 November 2007; Revised: 21 October 2008; Published: 2010
First available in Project Euclid: 31 January 2017

zbMATH: 1207.11097
MathSciNet: MR2600432
Digital Object Identifier: 10.1007/s11511-010-0044-9

Rights: 2010 © Institut Mittag-Leffler

Vol.204 • No. 1 • 2010
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