LARGE BIAS FOR INTEGERS WITH PRIME FACTORS IN ARITHMETIC PROGRESSIONS

We prove asymptotic formulas for the number of integers at most x that can be written as the product of k (>= 2) distinct primes P-l ... P-k with each prime factor in an arithmetic progression p(j) equivalent to a(j) mod q, (a(j), q) = 1 (q >= 3, 1 <= j <= k). For any A > 0, our result is uniform for 2 <= k <= A log log x. Moreover, we show that there are large biases toward certain arithmetic progressions (a(1) mod q, ..., a(k) mod q), and such biases have connections with Mertens' theorem and the least prime in arithmetic progressions.