Large bias for integers with prime factors in arithmetic progressions

TL;DR

This paper establishes an asymptotic count for integers with prime factors in specified arithmetic progressions and reveals significant biases in their distribution, linked to classical number theory conjectures and theorems.

Contribution

It provides a uniform asymptotic formula for the count of such integers and uncovers large biases in their distribution across arithmetic progressions.

Findings

Asymptotic formula for integers with prime factors in progressions

Identification of large biases in distribution

Connections to Mertens' theorem and least primes in progressions

Abstract

We prove an asymptotic formula for the number of integers $\leq x$ which can be written as the product of $k ~(\geq 2)$ distinct primes $p_1\cdots p_k$ 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 $A>0$, our result is uniform for $2\leq k\leq A\log\log x$. Moreover, we show that, there are large biases toward certain arithmetic progressions $(a_1 \bmod q, \cdots, a_k \bmod q)$, and such biases have connections with Mertens' theorem and the least prime in arithmetic progressions.