Large Bias for Integers With Prime Factors in Arithmetic Progressions

Abstract

We prove an asymptotic formula for the number of integers ≤ x which can be written as the product of k (≥ 2) distinct primes p 1 · · · p k with each prime factor in an arithmetic progression p j ≡ 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.

On Erdős and Sárközy’s sequences with Property P

“…Very recently Meng [14] used tools from analytic number theory to prove a generalization of this result to square-free integers having k prime factors in prescribed residue classes. The following is contained as a special case in [14, Lemma 9]:…”

“…The authors thank these institutions for their hospitality. The authors are also grateful to the referee for suggestions on the manuscript and would like to thank Xianchang Meng for some discussion on his recent paper [14].…”