Chebyshev's bias for products of $k$ primes

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.