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

Dixit, Anup

doi:10.1090/proc/14793

On Euler-Kronecker constants and the generalized Brauer-Siegel conjecture
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Proc. Amer. Math. Soc. 148 (2020), 1399-1414 Request permission

Abstract:

As a natural generalization of the Euler-Mascheroni constant $\gamma$, 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.

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