Published online by Cambridge University Press: 15 February 2018
We prove asymptotic formulas for the number of integers at most $x$ that can be written as the product of
$k~({\geqslant}2)$ distinct primes
$p_{1}\cdots p_{k}$ with each prime factor in an arithmetic progression
$p_{j}\equiv a_{j}\hspace{0.2em}{\rm mod}\hspace{0.2em}q$,
$(a_{j},q)=1$
$(q\geqslant 3,1\leqslant j\leqslant k)$. For any
$A>0$, our result is uniform for
$2\leqslant k\leqslant A\log \log x$. Moreover, we show that there are large biases toward certain arithmetic progressions
$(a_{1}\hspace{0.2em}{\rm mod}\hspace{0.2em}q,\ldots ,a_{k}\hspace{0.2em}{\rm mod}\hspace{0.2em}q)$, and such biases have connections with Mertens’ theorem and the least prime in arithmetic progressions.