# Zeta Functions

## September 18 - 22, 2006, Moscow, Russia

### Laboratoire J.-V. Poncelet

General

Announcement

Participants

Practical details

Program

Alexey Zykin

Independent University of Moscow, Russia

## The Generalized Brauer-Siegel Theorem

The classical Brauer-Siegel theorem states that if $k$ runs through the sequence of normal extensions of $\mathbb{Q}$ such that $n_k/\log|D_k|\to 0,$ 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 Brauer-Siegel 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 Brauer-Siegel theorem both in the number field and in the function field cases giving an overview of known results and open problems.