Large bias for integers with prime factors in arithmetic progressions

Xianchang Meng

Published 2016-07-07Version 1

We study the distribution of the product of $k ~(\geq 2)$ distinct primes $p_1\cdots p_k\leq x$ with each prime factor in an arithmetic progression $p_j\equiv a_j \bmod q$, $(a_j, q)=1$ $(q \geq 3, 1\leq j\leq k)$. For any $A>0$, we prove an asymptotic formula uniformly for $2\leq k\leq A\log\log x$ for the number of such integers. Moreover, we show that, there are large biases towards some certain arithmetic progressions $\boldsymbol{a}:=(a_1, \cdots, a_k)$, and such biases have connections with Mertens theorem and the least prime in arithmetic progressions.