Université de Montréal and University of Oxford
Small gaps between primes
It is believed that there should be infinitely many pairs of primes which differ by 2; this is the famous twin prime conjecture. More generally, it is believed that for every positive integer m there should be infinitely many sets of m primes with each set contained in an interval of size roughly m log m. Although proving these conjectures seems to be beyond our current techniques, recent progress has enabled us to obtain some partial results. We will introduce a refinement of the “GPY sieve method” for studying these problems. This refinement will allow us to show (amongst other things) that liminfn (pn+m – pn) < ∞ for any integer m, and so there are infinitely many bounded length intervals containing m primes. We will also discuss some extensions of this result.