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

Alexey Zykin Independent University of Moscow, Russia The Generalized BrauerSiegel TheoremThe classical BrauerSiegel theorem states that if $k$ runs through the sequence of normal extensions of $\mathbb{Q}$ such that $n_k/\logD_k\to 0,$ then $\log h_k R_k/\log \sqrt{D_k}\to 1.$ In this talk we will give a brief survey of what is known about the generalizations of this theorem. First, we will discuss the versions of the BrauerSiegel theorem where the conditions on the family of number fields are considerably weakened. Second, we will mention some explicit versions of the theorem due to P. Lebacque. Third, we will dwell on the higher dimensional analogues of the BrauerSiegel theorem both in the number field and in the function field cases giving an overview of known results and open problems. Go to the Laboratoire Poncelet home page. 

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