The generalized Brauer–Siegel conjecture for asymptotically exact families

Let $K/\mathbb{Q}$ be a number field, let $g_K=\log\sqrt{|d_K|}$ , and let $\mathcal{K}=\{K_i\}_{i\in\mathbb{N}}$ be an asymptotically exact family: the limits

\phi_q=\lim_{i\to\infty}\frac{N_q(K_i)}{g_{K_i}},\qquad \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}}

exist for every prime power $q$ , where $N_q(K_i)$ counts the non-archimedean places of norm $q$ , and $r_1(K_i),r_2(K_i)$ count the real and complex embeddings. Define

BS(\mathcal{K})=\lim_{i\to\infty}\frac{\log h_{K_i}R_{K_i}}{g_{K_i}},\qquad \rho(\mathcal{K})=\lim_{i\to\infty}\frac{\log\rho_{K_i}}{g_{K_i}}.

Tsfasman–Vlăduț conjecture. For every asymptotically exact family $\mathcal{K}$ ,

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, by the class number formula,

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

The conjecture generalizes the Brauer–Siegel theorem to families of number fields. Under the generalized Riemann hypothesis, the existence of these limits is known for every asymptotically exact family, but the asserted formulas are not established in general.

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).

The generalized Brauer–Siegel conjecture for asymptotically exact families

The generalized Brauer–Siegel conjecture for asymptotically exact families

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