Small gaps between primes

@article{Maynard2013SmallGB,
  title={Small gaps between primes},
  author={James Maynard},
  journal={Annals of Mathematics},
  year={2013},
  volume={181},
  pages={383-413}
}
  • J. Maynard
  • Published 19 November 2013
  • Mathematics
  • Annals of Mathematics
We introduce a renement of the GPY sieve method for studying prime k-tuples and small gaps between primes. This renement avoids previous limitations of the method and allows us to show that for each k, the prime k-tuples conjecture holds for a positive proportion of admissible k-tuples. In particular, lim infn(pn+m pn) <1 for every integer m. We also show that lim inf(pn+1 pn) 600 and, if we assume the Elliott-Halberstam conjecture, that lim infn(pn+1 pn) 12 and lim infn(pn+2 pn) 600. 
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