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. ...