Chebyshev's bias for products of $k$ primes

Xianchang Meng

Published 2016-06-15Version 1

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 and determine the bias between them, where $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity and $\omega(n)$ is the number of distinct prime factors of $n$. Under some reasonable assumptions, Rubinstein and Sarnak studied this problem for primes in arithmetic progressions (i.e. $\Omega(n)=1$); Ford and Sneed recently proved similar results for products of two primes with $\Omega(n)=2$ in arithmetic progressions. Under the same assumptions, we solve the Chebyshev's bias problem for products of $k$ primes for all $k\geq 1$ and for both cases of $\Omega(n)=k$ and $\omega(n)=k$.