Large bias for integers with prime factors in arithmetic progressions

Abstract

We prove an asymptotic formula for the number of integers $\le x$ which can be written as the product of $k(\ge 2)$ distinct primes $p_1\cdots p_k$ with each prime factor in an arithmetic progression $p_j\equiv a_j\mathrm{mod}q$, $(a_j,q)=1$ $(q\ge 3,1\le j\le k)$. For any $A>0$, our result is uniform for $2\le k\le A\log \log x$. Moreover, we show that, there are large biases toward certain arithmetic progressions $(a_1\mathrm{mod}q,\cdots ,a_k\mathrm{mod}q)$, and such biases have connections with Mertens' theorem and the least prime in arithmetic progressions.

13 pages. Comments are welcome

Cited by in corpus (3)

• Chebyshev's bias for products of irreducible polynomials
• An annotated bibliography for comparative prime number theory
• Biases in prime factorizations and Liouville functions for arithmetic progressions