Tsfasman–Vlăduț generalized Brauer–Siegel conjecture

Let $\mathcal{K}=\{K_i\}_{i\in\mathbb{N}}$ be an asymptotically exact family of number fields. For each $i$, let $h_{K_i}$ be the class number, $R_{K_i}$ the regulator, $g_{K_i}=\log\sqrt{d_{K_i}}$ the genus, and let $\phi_{\mathbb{R}}$, $\phi_{\mathbb{C}}$, and $\phi_q$ be the limits of the normalized numbers of real embeddings, complex embeddings, and non-archimedean places of norm $q$, respectively. Tsfasman–Vlăduț generalized Brauer–Siegel conjecture. The limit

$$BS(\mathcal{K})=\lim_{i\to\infty}\frac{\log h_{K_i}R_{K_i}}{g_{K_i}}$$

exists and equals

$$BS(\mathcal{K})=1+\sum_q\phi_q\log\frac{q}{q-1}-\phi_{\mathbb{R}}\log 2-\phi_{\mathbb{C}}\log 2\pi.$$

Equivalently, if $\rho_{K_i}$ denotes the residue of $\zeta_{K_i}(s)$ at $s=1$, then the limit

$$\rho(\mathcal{K})=\lim_{i\to\infty}\frac{\log\rho_{K_i}}{g_{K_i}}$$

exists and satisfies

$$\rho(\mathcal{K})=\sum_q\phi_q\log\frac{q}{q-1}.$$

The paper states that this conjecture is proved for asymptotically good towers and asymptotically bad families of number fields with solvable Galois closure; beyond those cases, the general assertion remains open.