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}
}
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. 
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References

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