Date: Thu, 13 Mar 2003 23:11:22 -0500 Reply-To: Hugh Montgomery <hlm@UMICH.EDU> Sender: Number Theory List <NMBRTHRY@LISTSERV.NODAK.EDU> From: Hugh Montgomery <hlm@UMICH.EDU> Subject: small gaps between primes To: NMBRTHRY@LISTSERV.NODAK.EDU Oberwolfach 13 March, 2003 This week at Oberwolfach, a meeting on Elementary and Analytic Number Theory is taking place, organized by Bruedern, Montgomery, and Vaughan. At this meeting, today, Dan Goldston spoke about his new joint work with Yildirim concerning primes in short intervals. They seem to have proved that liminf (p_{n+1} - p_n)/log p_n = 0. Indeed, it looks like they will be able to show that p_{n+1} - p_n is infinitely often as small as (log p_n)^{4/5}. It may be that the method can be applied also to long gaps between primes, with the prospect of possibly breaking the Erdos--Rankin order of magnitude. I would say that this is the biggest excietement that prime number theory has seen since the Bombieri--Vinogradov theorem was proved in 1966. Hugh Montgomery