# 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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