Small gaps between primes

By James Maynard. Graduate Studies in Mathematics, 2013

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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.

Subfields: Algebra and Number Theory, Geometry and Topology, Theoretical Computer Science

View Paper: https://www.semanticscholar.org/paper/9511d9965a4a0b9a3f21997a43c8ce07d9f11562