Tsfasman–Vladuts generalized Brauer–Siegel conjecture

Let $(K_i)_{i\in\mathbb{N}}$ be an asymptotically exact family of number fields. For each prime power $q$, let $\phi_q$ denote the corresponding asymptotic parameter, and let $\phi_{\mathbb{R}}$ and $\phi_{\mathbb{C}}$ denote the asymptotic parameters for real and complex places. Write $h(K_i)$ for the class number, $R(K_i)$ for the regulator, and $\Delta(K_i)$ for the discriminant.

Tsfasman–Vladuts conjecture. One has

$$\lim_{i\to\infty}\frac{\log\big(h(K_i) \, R(K_i)\big)}{\log\sqrt{|\Delta(K_i)|}} = 1+\sum_q \phi_q\log\frac{q}{q-1}-\phi_{\mathbb{R}}\log 2-\phi_{\mathbb{C}}\log (2\pi),$$

where the sum on the right runs over all prime powers. This conjecture extends the classical Brauer–Siegel theorem by removing the bounded-degree hypothesis and describes the asymptotic behavior of class numbers and regulators in asymptotically exact families; the notation and convergence hypotheses are part of the Tsfasman–Vladuts framework.