# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.332011

Acta Mathematica, 204, (1), 1-47, 2007.11
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$n$th 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.
@inproceedings{daniel2007primes,
title={Primes in tuples II},
author={Daniel A. Goldston, János Pintz, and Cem Yalçin Yıldırım},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203354597948720},
booktitle={Acta Mathematica},
volume={204},
number={1},
pages={1-47},
year={2007},
}

Daniel A. Goldston, János Pintz, and Cem Yalçin Yıldırım. Primes in tuples II. 2007. Vol. 204. In Acta Mathematica. pp.1-47. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203354597948720.