Chebyshev's bias for products of irreducible polynomials

Journal article, 2021

For any k≥1, this paper studies the number of polynomials having k irreducible factors (counted with or without multiplicities) in Fq[t] among different arithmetic progressions. We obtain asymptotic formulas for the difference of counting functions uniformly for k in a certain range. In the generic case, the bias dissipates as the degree of the modulus or k gets large, but there are cases when the bias is extreme. In contrast to the case of products of k prime numbers, we show the existence of complete biases in the function field setting, that is the difference function may have constant sign. Several examples illustrate this new phenomenon.