Chebyshev's bias for products of k primes

For any k >= 1, we study the distribution of the difference between the number of integers n <= 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 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 (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.