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

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 Vlǎduţ. 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.

16 pages, to appear in the Proceedings of the American Mathematical Society