Small gaps between primes exist

@inproceedings{Goldston2005SmallGB,
  title={Small gaps between primes exist},
  author={D. A. Goldston and Yoichi Motohashi and Janos Pintz and C. Y. Yildirim},
  year={2005}
}
In the recent preprint (3), Goldston, Pintz, and Yoldorom established, among other things, (0) liminf n→∞ pn+1 − pn log pn = 0, with pn the nth prime. In the present article, which is essentially self-contained, we shall develop a simplified account of the method used in (3). While (3) also includes quantitative versions of (0), we are concerned here solely with proving the qualitative (0), which still exhibits all the essentials of the method. We also show here that an improvement of the… Expand
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TLDR
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PRIMES IN TUPLES I 1 Primes in Tuples I
We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming theExpand
Are there arbitrarily long arithmetic progressions in the sequence of twin primes? II
In an earlier work it was shown that the Elliott-Halberstam conjecture implies the existence of infinitely many gaps of size at most 16 between consecutive primes. In the present work we show thatExpand
Primes in tuples I
We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming theExpand
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References

SHOWING 1-4 OF 4 REFERENCES
SMALL GAPS BETWEEN PRIMES II ( PRELIMINARY )
We examine an idea for approximating prime tuples. 1. Statement of results (Preliminary) In the present work we will prove the following result. Let pn denote the nth prime. Then (1.1) lim inf n→∞Expand
On the distribution of primes in short intervals
One of the formulations of the prime number theorem is the statement that the number of primes in an interval ( n , n + h ], averaged over n ≤ N , tends to the limit λ, when N and h tend to infinityExpand
Le grand crible dans la théorie analytique des nombres
© Société mathématique de France, 1974, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec lesExpand
Titchmarsh, The theory of the Riemann zetafunction
  • 1951