The generalized Brauer–Siegel conjecture for asymptotically exact families

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The generalized Brauer–Siegel conjecture for asymptotically exact families

   Let $\mathcal{K}=\{K_i\}$ be a family of number fields. Write $g_{K_i}=\log\sqrt{d_{K_i}}$, let $r_1(K_i)$ and $r_2(K_i)$ be the numbers of real and complex embeddings, and let $N_q(K_i)$ be the number of ideals of $\mathcal{O}_{K_i}$ with norm $q$. Suppose the limits

   $$\phi_{\mathbb{R}}=\lim_{i\to\infty}\frac{r_1(K_i)}{g_{K_i}},\qquad \phi_{\mathbb{C}}=\lim_{i\to\infty}\frac{r_2(K_i)}{g_{K_i}},\qquad \phi_q=\lim_{i\to\infty}\frac{N_q(K_i)}{g_{K_i}}$$

   exist for every prime power $q$, so that $\mathcal{K}$ is asymptotically exact. Let $\rho_{K_i}$ be the residue of $\zeta_{K_i}(s)$ at $s=1$.

   Generalized Brauer–Siegel conjecture. Then

   $$\lim_{i\to\infty}\frac{\log h_{K_i}R_{K_i}}{g_{K_i}}=1+\sum_q\phi_q\log\frac{q}{q-1}-\phi_{\mathbb{R}}\log 2-\phi_{\mathbb{C}}\log(2\pi).$$

   Equivalently,

   $$\lim_{i\to\infty}\frac{\log\rho_{K_i}}{g_{K_i}}=\sum_q\phi_q\log\frac{q}{q-1}.$$

   This is the generalized form formulated by Tsfasman and Vlăduţ; the paper discusses establishing it for asymptotically good towers over families of almost $S_n$-fields.

   source: Anup B Dixit, “Brauer-Siegel theorem for families of number fields over almost Sn fields”, arXiv:2601.18408 (2026).