Chebyshev's bias for products of $k$ primes

Abstract

For any $k\ge 1$, we study the distribution of the difference between the number of integers $n\le x$ with $\omega \left(n\right)=k$ or $\mathrm{\Omega }\left(n\right)=k$ in two different arithmetic progressions, where $\omega \left(n\right)$ is the number of distinct prime factors of $n$ and $\mathrm{\Omega }\left(n\right)$ 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 $\mathrm{\Omega }\left(n\right)=k$ have preference for quadratic nonresidue 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 $\omega \left(n\right)=k$ always have preference for quadratic residue classes. Moreover, as $k$ increases, the biases become smaller and smaller for both of the two cases.