The signs of the Stieltjes constants associated with the Dedekind zeta function

TL;DR

This paper explores the properties and signs of the Stieltjes constants associated with the Dedekind zeta function of a number field, extending classical results and providing new insights into their behavior.

Contribution

It establishes a new expression for the Stieltjes constants of number fields and analyzes their signs, extending classical results from the rational case to general number fields.

Findings

Derived a formula for $\gamma_n(K)$ similar to the classical case.

Analyzed the signs of $\gamma_n(K)$ for various number fields.

Provided insights into the behavior of these constants in algebraic number theory.

Abstract

The Stieltjes constants $\gamma_n(K)$ of a number field $K$ are the coefficients of the Laurent expansion of the Dedekind zeta function $\zeta_K(s)$ at its pole $s=1$. In this paper, we establish a similar expression of $\gamma_n(K)$ as Stieltjes obtained in 1885 for $\gamma_n(\mathbb Q)$. We also study the signs of $\gamma_n(K)$.