Zeta Functions

September 18 - 22, 2006, Moscow, Russia

Laboratoire J.-V. Poncelet

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Philippe Lebacque

Institut de Mathematiques de Luminy, France

Generalized Mertens and Brauer-Siegel theorems.

The Tsfasman-Vladuts' generalized Brauer Siegel Theorem gives us the asymptotic behaviour of the residue of zeta at $s=1$ in a tower of fields. It's closely related to Mertens theorem which can be seen as the finite step of Brauer-Siegel theorem in the case of $\mathbb{Q}$. Mireille Car generalized it in the case of function fields, but it can also be generalized in the case of any global fields, and this leads to an explicite version of B-S theorem under GRH. Finally, it also can be generalized in the case of projective smooth variety over a finite field and this gives us an effective version of a result of Zykin about the residue of zeta in $s=d$.