Small gaps between primes or almost primes

Abstract

Let $p_n$ denote the $n^{th}$ prime. Goldston, Pintz, and Yildirim recently proved that ${\mathrm{liminf}}_{n\to \infty }\frac{(p_{n+1}-p_n)}{\log p_n}=0.$ We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let $q_n$ denote the $n^{th}$ number that is a product of exactly two distinct primes. We prove that ${\mathrm{liminf}}_{n\to \infty }(q_{n+1}-q_n)\le 26.$ If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6.

49 pages