# Primes in tuples II

@article{Goldston2007PrimesIT,
title={Primes in tuples II},
author={D. A. Goldston and Janos Pintz and Cem Yalçin Yıldırım},
journal={Acta Mathematica},
year={2007},
volume={204},
pages={1-47}
}
• Published 15 October 2007
• Mathematics
• Acta Mathematica
AbstractWe 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+1−pn 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.
A generalization of the Goldston–Pintz–Yildirim prime gaps result to number fields
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