Under two assumptions, we determine the distribution of the difference between two functions each counting the numbers less than or equal to $x$ that are in a given arithmetic progression modulo $q$ and the product of two primes. The two assumptions are (i) the extended Riemann hypothesis for Dirichlet $L$-functions modulo $q$, and (ii) that the imaginary parts of the nontrivial zeros of these $L$-functions are linearly independent over the rationals. Our results are analogues of similar results proved for primes in arithmetic progressions by Rubinstein and Sarnak.
"Chebyshev's Bias for Products of Two Primes." Experiment. Math. 19 (4) 385 - 398, 2010.