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### Small gaps between primes exist

Daniel Alan Goldston, Yoichi Motohashi, János Pintz, and Cem Yalçın Yıldırım
Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 82, Number 4 (2006), 61-65.

#### Abstract

In the preprint [3], Goldston, Pintz, and Yıldırım established, among other things, $$\liminf_{n\to\infty}{p_{n+1}-p_n\over\log p_n}=0,$$ with $p_n$ the $n$th prime. In the present article, which is essentially self-contained, we shall develop a simplified account of the method used in [3]. We include a short expository last section.

First Page:
Primary Subjects: 11N05
Secondary Subjects: 11P32
Keywords: Prime number
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.pja/1146576181
Digital Object Identifier: doi:10.3792/pjaa.82.61
Zentralblatt MATH identifier: 05123005
Mathematical Reviews number (MathSciNet): MR2222213

### References

E. Bombieri, Le grand crible dans la théorie analytique des nombres, second édition revue et augmentée, Astérisque, 18, Soc. Math. France, Paris, 1987.
Mathematical Reviews (MathSciNet): MR891718
P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), no. 1, 4--9; Corrigendum, ibid, 28 (1981), 86.
Mathematical Reviews (MathSciNet): MR409385
D. A. Goldston, J. Pintz, and C. Y. Y\ild\ir\im, Small gaps between primes II (Preliminary). (February 8, 2005). See also [2005-19 of http://aimath.org/preprints.html].
E. C. Titchmarsh, The theory of the Riemann zeta-function, Clarendon Press, Oxford, 1951.
Mathematical Reviews (MathSciNet): MR46485
Zentralblatt MATH: 0042.07901
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