On the least almost-prime in an arithmetic progression

Abstract

For all sufficiently large $$, in any arithmetic progression $$ in which $$ and $$ are relatively prime there exists a positive integer with at most two prime factors (counted with multiplicity) which is asymptotically less than $$. The proof uses the weighted sieve of Greaves–Halberstam–Richert with bilinear remainder terms and Selberg's sieve.