Primes in tuples II

Daniel A. Goldston Department of Mathematics, San José State University János Pintz Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences Cem Yalçin Yıldırım Department of Mathematics, Bog̃aziçi University

TBD mathscidoc: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.
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@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.
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