Order of Zeros of Dedekind Zeta Functions

TL;DR

This paper proves that Dedekind zeta functions of certain number fields have infinitely many zeros of multiplicity at least 2 or 3, based on the Galois group structure, confirming predictions of the Artin holomorphy conjecture.

Contribution

It provides an unconditional proof that Dedekind zeta functions have multiple zeros under specific Galois extension conditions, extending previous results.

Findings

Infinitely many zeros of multiplicity ≥ 2 when $L/K$ is nonabelian Galois extension.

Zeros of order 3 when $ ext{Gal}(L/K)$ has an irreducible representation of degree ≥ 3.

Supports predictions of the Artin holomorphy conjecture.

Abstract

Answering a question of Browkin, we provide a new unconditional proof that the Dedekind zeta function of a number field $L$ has infinitely many nontrivial zeros of multiplicity at least 2 if $L$ has a subfield $K$ for which $L/K$ is a nonabelian Galois extension. We also extend this to zeros of order 3 when $\operatorname{Gal}(L/K)$ has an irreducible representation of degree at least 3, as predicted by the Artin holomorphy conjecture.