Zeta FunctionsSeptember 18  22, 2006, Moscow, RussiaLaboratoire J.V. Poncelet 

Philippe Lebacque Institut de Mathematiques de Luminy, France Generalized Mertens and BrauerSiegel theorems.The TsfasmanVladuts' 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 BrauerSiegel 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 BS 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 $s=d$. Go to the Laboratoire Poncelet home page. 

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