Bounded gaps between products of primes with applications to ideal class groups and elliptic curves
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA
Correspondence: Correspondence to be sent to: thorne{at}math.wisc.edu
In their recent papers, Goldston, Graham, Pintz, and Y
ldrm [12, 13] use a variant of the Selberg sieve to prove the existence of small gaps between E2 numbers, that is, square-free numbers with exactly two prime factors. We apply their techniques to prove similar bounds for Er numbers for any r 
2, where these numbers are required to have all of their prime factors in a set of primes 
. Our result holds for any 
of positive density that satisfies a Siegel–Walfisz condition regarding distribution in arithmetic progressions. We also prove a stronger result in the case that 
satisfies a Bombieri–Vinogradov condition. We were motivated to prove these generalizations because of recent results of Ono [22] and Soundararajan [25]. These generalizations yield applications to divisibility of class numbers, nonvanishing of critical values of L-functions, and triviality of ranks of elliptic curves.