Chebyshev's Bias for Products of Two Primes

Abstract

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.