Numbers with at most $2$ prime factors in short arithmetic progressions

Abstract

We show that if $\frac{L}{\phi (q)\log X}\to \infty$ as $X\to \infty$, almost all $(a,x)\in (\mathbb{Z}/q\mathbb{Z}{)}^{\times }\times [X,2X]$ are such that there exists a product of at most two primes in $[x,x+L]$ congruent to $a\mathrm{mod}q$.

15 pages