Primes in the Chebotarev density theorem for all number fields

Abstract

We establish an explicit bound for the least prime occurring in the Chebotarev density theorem without any restriction. Let $L/K$ be any Galois extension of number fields such that $L\not=\mathbb{Q}$, and let $C$ be a conjugacy class in the Galois group of $L/K$. We show that there exists an unramified prime $\mathfrak{p}$ of $K$ such that $\sigma_{\mathfrak{p}}=C$ and $N \mathfrak{p} \le d_{L}^{B}$ with $B= 310$. This improves the value $B=12\,577$ as proven by Ahn and Kwon. In comparison to previous works on the subject, we modify the weights to detect the least prime, and we use a new version of Tur\'an's power sum method which gives a stronger Deuring-Heilbronn (zero-repulsion) phenomenon. In addition, we refine the analysis of how the location of the potential exceptional zero for $\zeta_L(s)$ affects the final result. We also use Fiori's numerical verification for $L$ up to a certain discriminant height. Finally, we provide a lower bound for the number of unramified primes $\mathfrak{p}$ of $K$ such that $\sigma_{\mathfrak{p}}=C$.