Higher Euler-Kronecker Constants of Number fields

TL;DR

This paper investigates the higher Euler-Kronecker constants of number fields, providing arithmetic formulas and bounds, and extending previous results by Ihara to better understand their arithmetic significance.

Contribution

It introduces new arithmetic formulas and bounds for higher Euler-Kronecker constants, generalizing prior results and deepening understanding of their properties.

Findings

Derived explicit arithmetic formulas for the constants

Established bounds for the higher Euler-Kronecker constants

Extended results of Ihara to broader contexts

Abstract

The higher Euler-Kronecker constants of a number field $K$ are the coefficients in the Laurent series expansion of the logarithmic derivative of the Dedekind zeta function about $s=1$. These coefficients are mysterious and seem to contain a lot of arithmetic information. In this article, we study these coefficients. We prove arithmetic formulas satisfied by them and prove bounds. We generalize certain results of Ihara.