2005/05/14 by D. A. Goldston, Y. Motohashi, Goldston, D. A. +5
Mathematics · #11N05 (Primary) #11P32 (Secondary) #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11N05 #msc:11P32
paper · pdf · doi:10.48550/arxiv.math/0505300
8 pages
arxiv created 2005/05/14 · arxiv updated 2009/12/01
In the recent preprint [3], Goldston, Pintz, and Yıldırım established, among other things, \liminfn→∞pn+1-pn\overlog pn=0,\leqno(0) with pn the nth prime. In the present article, which is essentially self-contained, we shall develop a simplified account of the method used in [3]. While [3] also includes quantitative versions of (0), we are concerned here solely with proving the qualitative (0), which still exhibits all the essentials of the method. We also show here that an improvement of the Bombieri--Vinogradov prime number theorem would give rise infinitely often to bounded differences between consecutive primes. We include a short expository last section. Detailed discussions of quantitative results and a historical review will appear in the publication version of [3] and its continuations.