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.

Keywords:

• Prime number race
• Chebyshev's bias