Extreme values of Artin $L$-functions at $s=1$

Abstract

Let $K/\mathbb{Q}$ be a Galois extension of number fields with Galois group $G$. Let $χ$ be any character on $G$ and $κ(χ)$ be the leading coefficient of the Laurent series of the Artin $L$-function $L(s,χ)$ at $s=1$. Cho, Lemke Oliver, and Zaman (2026) recently established a conditional bound for $κ(χ)$ assuming GRH for the Dedekind zeta function of $K$. Up to implied constants, we show that the extremal values of these bounds are attained for an infinite family of extensions $K/\mathbb{Q}$ and all irreducible faithful characters $χ$, provided $G$ is a regular Galois group over $\mathbb{Q}$. All characters $χ$ are permitted without restriction when $G$ is abelian, $A_n, S_n$, or any sporadic group. Our method removes the assumptions of GRH and Artin's holomorphy conjecture from an argument of Duke (2003) by following the strategy of Lemke Oliver, Thorner, and Zaman (2024) on the existence of extremal class numbers for number fields of arbitrary degree. Key ingredients include progress by Débes (2017) towards Malle's conjecture, and an Artin $L$-function large sieve by Lemke Oliver and Smith (2024).