Ties in Function Field Prime Races

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 $$\ddot{I}€(N;m,c_1)=\ddot{I}€(N;m,c_2)=\cdots =\ddot{I}€(N;m,c_k)$$ 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 Möbius 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.

24 pages, 6 tables