Small gaps between primes
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.