Inequities in the Shanks-Renyi prime number race over function fields

TL;DR

This paper investigates biases and asymptotic behaviors in prime number races over function fields, extending classical number field results and revealing new phenomena when the Linear Independence hypothesis fails.

Contribution

It provides an asymptotic formula for prime number race densities over function fields and analyzes how these biases differ from number fields, especially when LI does not hold.

Findings

Convergence rate of race densities to 1/2 as degree of m grows

Different behaviors in two-way versus multi-way races

Existence of races with vanishing densities when LI is false

Abstract

Fix a prime $p >2$ and a finite field $\mathbb{F}_{q}$ with $q$ elements, where $q$ is a power of $p$. Let $m$ be a monic polynomial in the polynomial ring $\mathbb{F}_{q}[T]$ such that $deg(m)$ is large. Fix an integer $r\geq 2$, and let $a_1,\dots,a_r$ be distinct residue classes modulo $m$ that are relatively prime to $m$. In this paper, we derive an asymptotic formula for the natural density $\delta_{m;a_1,\dots,a_r}$ of the set of all positive integers $X$ such that $\sum\limits_{N=1}^{X} \pi_{q}(a_1,m,N) > \sum\limits_{N=1}^{X} \pi_{q}(a_2,m,N) > \dots > \sum\limits_{N=1}^{X} \pi_{q}(a_r,m,N)$, where $\pi_{q}(a_i,m,N)$ denotes the number of irreducible monic polynomials in $\mathbb{F}_{q}[T]$ of degree $N$ that are congruent to $ a_i \bmod m$, under the assumption of LI (Linear Independence Hypothesis). Many consequences follow from our results. First, we deduce the exact rate at which $\delta_{m;a_1,a_2}$ converges to $\frac{1}{2}$ as $deg(m)$ grows, where $a_1$ is a quadratic non-residue and $a_2$ is a quadratic residue modulo $m$, generalizing the work of Fiorilli and Martin. Furthermore, similarly to the number field setting, we show that two-way races behave differently than races involving three or more competitors, once $deg(m)$ is large. In particular, biases do appear in races involving three or more quadratic residues (or quadratic non-residues) modulo $m$. This work is a function field analog of the work of Lamzouri, who established similar results in the number field case. However, we exhibit some examples of races in function fields where LI is false, and where the associated densities vanish, or behave differently than in the number field setting.