Chebyshev's bias for products of $k$ primes

TL;DR

This paper investigates the distribution biases of integers with a fixed number of prime factors in different arithmetic progressions, confirming a conjecture and revealing how biases diminish as the number of prime factors increases.

Contribution

It establishes the bias patterns for integers with fixed prime factors, confirming Hudson's conjecture and analyzing how biases change with increasing prime factors.

Findings

Integers with $ ext{Ω}(n)=k$ prefer quadratic non-residue classes if $k$ is odd.

Integers with $ ext{Ω}(n)=k$ prefer quadratic residue classes if $k$ is even.

Biases decrease as the number of prime factors $k$ increases.

Abstract

For any $k\geq 1$, we study the distribution of the difference between the number of integers $n\leq x$ with $\omega(n)=k$ or $\Omega(n)=k$ in two different arithmetic progressions, where $\omega(n)$ is the number of distinct prime factors of $n$ and $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity . Under some reasonable assumptions, we show that, if $k$ is odd, the integers with $\Omega(n)=k$ have preference for quadratic non-residue classes; and if $k$ is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Richard Hudson. However, the integers with $\omega(n)=k$ always have preference for quadratic residue classes. Moreover, as $k$ increases, the biases become smaller and smaller for both of the two cases.