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

Abstract

Fix a prime p > 2 $p >2$ and a finite field F q $\mathbb {F}_{q}$ with q elements, where q is a power of p. Let m be a monic polynomial in the polynomial ring F q [ T ] $\mathbb {F}_{q}[T]$ such that deg ( m ) $\deg (m)$ is large. Fix an integer r ⩾ 2 $r\geqslant 2$ , and let a 1 , ⋯ , a r $a_1,\dots ,a_r$ be distinct residue classes modulo m that are relatively prime to m. In this paper, we derive an asymptotic formula for the natural density δ m ; a 1 , ⋯ , a r $\delta _{m;a_1,\dots ,a_r}$ of the set of all positive integers X such that ∑ N = 1 X π q ( a 1 , m , N ) > ∑ N = 1 X π q ( a 2 , m , N ) > ⋯ > ∑ N = 1 X π q ( a r , m , N ) $\sum _{N=1}^{X} \pi _{q}(a_1,m,N) > \sum _{N=1}^{X} \pi _{q}(a_2,m,N)>\cdots > \sum _{N=1}^{X} \pi _{q}(a_r,m,N)$ , where π q ( a i , m , N ) $\pi _{q}(a_i,m,N)$ denotes the number of irreducible monic polynomials in F q [ T ] $\mathbb {F}_{q}[T]$ of degree N that are congruent to a i mod m $ 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 δ m ; a 1 , a 2 $\delta _{m;a_1,a_2}$ converges to 1 2 $\frac{1}{2}$ as deg ( m ) $\deg (m)$ grows, where a1 is a quadratic non-residue and a2 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 ) $\deg (m)$ 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.