Chebyshev's bias for products of two primes

TL;DR

This paper investigates the distribution of differences in counting functions for products of two primes versus primes in arithmetic progressions, revealing a reversed bias compared to primes alone, under certain hypotheses.

Contribution

It extends the analysis of Chebyshev's bias to products of two primes, showing a reversed bias compared to primes, under the Extended Riemann Hypothesis and linear independence assumptions.

Findings

Bias for products of two primes is reversed from that for primes.

Under assumptions, 89.4% of the time, products of two primes dominate in certain residue classes.

Results are analogous to those for primes, but with reversed bias.

Abstract

Under two assumptions, we determine the distribution of the difference between two functions each counting the numbers < 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 analogs of similar results proved for primes in arithmetic progressions by Rubinstein and Sarnak. In particular, we show that the bias for products of two primes is always reversed from the bias for primes. For example, while Rubinstein and Sarnak showed, under (i) and (ii), that 99.6% of the time there are more primes up to x which are 3 mod 4 than those which are 1 mod 4, we show that 89.4% of the time, there are more products of two primes which are 1 mod 4 than 3 mod 4.