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Asymptotic properties of Dedekind zeta functions in families of number fields

Alexey Zykin — 2010

Journal de Théorie des Nombres de Bordeaux

The main goal of this paper is to prove a formula that expresses the limit behaviour of Dedekind zeta functions for ℜ s > 1 / 2 in families of number fields, assuming that the Generalized Riemann Hypothesis holds. This result can be viewed as a generalization of the Brauer–Siegel theorem. As an application we obtain a limit formula for Euler–Kronecker constants in families of number fields.

On the number of rational points of Jacobians over finite fields

Philippe Lebacque;  Alexey Zykin — 2015

Acta Arithmetica

We prove lower and upper bounds for the class numbers of algebraic curves defined over finite fields. These bounds turn out to be better than most of the previously known bounds obtained using combinatorics. The methods used in the proof are essentially those from the explicit asymptotic theory of global fields. We thus provide a concrete application of effective results from the asymptotic theory of global fields and their zeta functions.

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