# On Euler-Kronecker constants and the generalized Brauer-Siegel conjecture

## TL;DR

This paper explores the relationship between Euler-Kronecker constants and the generalized Brauer-Siegel conjecture, showing how bounds on these constants influence conjecture validity and zero distribution of Dedekind zeta-functions.

## Contribution

It proves that bounds on Euler-Kronecker constants imply the generalized Brauer-Siegel conjecture for certain number field towers and refines zero estimates for Dedekind zeta-functions.

## Key findings

- Bounds on Euler-Kronecker constants imply the generalized Brauer-Siegel conjecture.
- Known bounds for cyclotomic fields lead to improved zero distribution estimates.
- The work connects constants, conjectures, and zero distributions in number theory.

## Abstract

As a natural generalization of the Euler-Mascheroni constant $\gamma$, Y. Ihara introduced the Euler-Kronecker constant $\gamma_K$ attached to any number field $K$. In this paper, we prove that a certain bound on $\gamma_K$ in a tower of number fields $\mathcal{K}$ implies the generalized Brauer-Siegel conjecture for $\mathcal{K}$ as formulated by Tsfasman and Vl\v{a}du\c{t}. Moreover, we use known bounds on $\gamma_K$ for cyclotomic fields to obtain a finer estimate for the number of zeros of the Dedekind zeta-function $\zeta_K(s)$ in the critical strip.