Ties in Function Field Prime Races

TL;DR

This paper investigates the phenomenon of ties in prime number races within function fields, providing explicit examples and two proofs involving L-functions and group actions.

Contribution

It introduces new explicit examples of ties in function field prime races and offers two distinct proofs, expanding understanding of Chebyshev's bias analogues.

Findings

Identifies infinitely many examples of ties under certain conditions.

Provides two different proofs: one via L-functions and M"obius inversion, another via group actions.

Includes characteristic 2 cases in the analysis.

Abstract

The function field analogue of Chebyshev's bias was first studied by Cha. In this paper, we study *ties* in this race, namely collections of distinct congruence classes $c_1, \dots, c_k \in (\mathbb{F}_q[T] / m)^\times$ for which $$\pi(N; m, c_1) = \pi(N; m, c_2) = \dots = \pi(N; m, c_k)$$ holds for infinitely many $N$. We provide infinitely many examples of $(m, c_1, \dots, c_k)$ for which the tie holds whenever $N$ satisfies certain congruence conditions. We give two different proofs: first, via the explicit formula for prime counts in terms of $L$-functions together with a matrix analogue of M\"obius inversion, where exceptional pairs of Galois-conjugate elements in the corresponding cyclotomic fields produce ties; and second, via an explicit bijection arising from the $\mathrm{GL}_2(\mathbb{F}_q)$-action. Our examples also include characteristic 2 cases.