## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Small gaps between primes exist

#### 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.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 82, Number 4 (2006), 61-65.

Dates
First available in Project Euclid: 2 May 2006

http://projecteuclid.org/euclid.pja/1146576181

Digital Object Identifier
doi:10.3792/pjaa.82.61

Mathematical Reviews number (MathSciNet)
MR2222213

Zentralblatt MATH identifier
05123005

Subjects
Primary: 11N05: Distribution of primes
Secondary: 11P32: Goldbach-type theorems; other additive questions involving primes

Keywords
Prime number

#### Citation

Goldston, Daniel Alan; Motohashi, Yoichi; Pintz, János; Yıldırım, Cem Yalçın. Small gaps between primes exist. Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 4, 61--65. doi:10.3792/pjaa.82.61. http://projecteuclid.org/euclid.pja/1146576181.

#### References

• E. Bombieri, Le grand crible dans la théorie analytique des nombres, second édition revue et augmentée, Astérisque, 18, Soc. Math. France, Paris, 1987.
• P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), no. 1, 4–9; Corrigendum, ibid, 28 (1981), 86.
• 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.