The Brauer-Siegel and Tsfasman–Vlǎdut̨ theorems for almost normal extensions of number fields

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 $\log h_k R_k/\log \sqrt{|D_k|}\to 1$. First, in this paper we obtain the generalization of the Brauer–Siegel and Tsfasman–Vlǎdut̨ theorems to the case of almost normal number fields. Second, using the approach of Hajir and Maire, we construct several new examples concerning the Brauer–Siegel ratio in asymptotically good towers of number fields. These examples give smaller values of the Brauer–Siegel ratio than those given by Tsfasman and Vlǎdut̨.