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arXiv · 1908.03044

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

Anup B. Dixit
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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.

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Anup B. Dixit. 2019-08-08. On Euler-Kronecker constants and the generalized Brauer-Siegel conjecture. https://arxiv.org/abs/1908.03044

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