Joint IAS/Princeton University Number Theory Seminar

We will introduce a refinement of the `GPY sieve method' for studying small gaps between primes. This refinement will allow us to show that \(\liminf_n(p_{n+m}-p_n) < \infty\) for any integer \(m\), and so there are infinitely many bounded length intervals containing \(m\) primes. Moreover, this method also applies to any subset of the primes which are reasonably well-distributed in arithmetic progressions. We will also discuss more recent developments from the Polymath project which improve the numerical bounds on \(\liminf_n(p_{n+1}-p_n)\).