A zero density estimate for Dedekind zeta functions

TL;DR

This paper establishes a zero density estimate for Dedekind zeta functions associated with Galois extensions, advancing number theory by removing reliance on unproven conjectures in related problems.

Contribution

It provides the first zero density estimate for Dedekind zeta functions that does not depend on the strong Artin conjecture.

Findings

Removed the need for the strong Artin conjecture in Chebotarev density theorem error estimates.

Extended zero density results to families of Dedekind zeta functions for Galois extensions.

Improved understanding of the distribution of zeros of Dedekind zeta functions.

Abstract

Given a nontrivial finite group $G$, we prove the first zero density estimate for families of Dedekind zeta functions associated to Galois extensions $K/\mathbb{Q}$ with $\mathrm{Gal}(K/\mathbb{Q})\cong G$ that does not rely on unproven progress towards the strong form of Artin's conjecture. We use this to remove the hypothesis of the strong Artin conjecture from the work of Pierce, Turnage-Butterbaugh, and Wood on the average error in the Chebotarev density theorem and $\ell$-torsion in ideal class groups.