Extreme residues of Dedekind zeta functions

TL;DR

This paper establishes precise bounds for the residues of Dedekind zeta functions across certain families of number fields, advancing understanding of their extremal behavior under specific conjectural assumptions.

Contribution

It provides the true upper and lower bounds of Dedekind zeta residues for $S_{d+1}$-fields, with new results for $S_5$-fields assuming the strong Artin conjecture.

Findings

Determined bounds for residues in $S_{d+1}$-fields for $d=2,3,4$

Established existence of infinite families achieving these bounds

Extended results to $S_5$-fields under conjectural assumptions

Abstract

In a family of $S_{d+1}$-fields ($d=2,3,4$), we obtain the true upper and lower bound of the residues of Dedekind zeta functions except for a density zero set. For $S_5$-fields, we need to assume the strong Artin conjecture. We also show that there exists an infinite family of number fields with the upper and lower bound, resp.