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

Abstract

Fix a prime p > 2 and a finite field Fq with q elements, where q is a power of p. Let m be a monic polynomial in the polynomial ring Fq[T ] such that deg(m) is large. Fix an integer r 2, and let a1, . . . , ar be distinct residue classes modulo m that are relatively prime to m. In this paper, we derive an asymptotic formula for the natural density δm;a 1 ,...,ar of the set of all positive integers X such thatwhere πq(ai, m, N ) denotes the number of irreducible monic polynomials in Fq[T ] of degree N that are congruent to ai mod m, under the assumption of LI (Linear Independence Hypothesis). Many consequences follow from our results. First, we deduce the exact rate at which δm;a 1 ,a 2 converges to 1 2 as 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) 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.