Chebyshev's bias for products of irreducible polynomials

TL;DR

This paper investigates the distribution of polynomials with a fixed number of irreducible factors in function fields, revealing phenomena like biases and their dissipation or persistence depending on parameters.

Contribution

It provides asymptotic formulas for counting polynomials with specified irreducible factors across arithmetic progressions, highlighting the existence of complete biases in the function field setting.

Findings

Bias dissipates as degree or number of factors increases

Complete biases can occur with constant sign in certain cases

Asymptotic formulas describe distribution of polynomials with fixed irreducible factors

Abstract

For any $k\geq 1$, this paper studies the number of polynomials having $k$ irreducible factors (counted with or without multiplicities) in $\mathbf{F}_q[t]$ 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.