Let denote the prime. Goldston,
Pintz, and Yıldırım
recently proved that
We give an alternative proof of this result. We also prove some
corresponding results for numbers with two prime factors. Let denote the number that is a product of
exactly two distinct primes. We prove
that
If an appropriate generalization of the Elliott-Halberstam
Conjecture is true, then the above bound can be improved to .