Year: 2026
Type: preprint
Abstract: Let $k$ be a number field. We prove that for every quadratic extension $F/k$, every real zero $β\in(0,1)$ of $L(s,χ_{F/k})$ satisfies $(1-β)\log d_F\ge c(k)$ with an effective constant $c(k)>0$, and we deduce (through a reduction of Stark) that the Brauer-Siegel theorem holds effectively and uniformly over all Galois extensions $K$ of $k$, in the form $h_K R_K \ge c(k)\, d_K^{1/2}/((2π)^{[K:\mathbb{Q}]}\log d_K)$. Stark's effective cases of the Brauer-Siegel theorem excluded the fields with quadratic subfields, since those subfields carried the ineffectivity of Siegel's theorem; over a base field $k$, the first result removes this obstruction. Both rely on a uniform bound for Chebotarev sums. Let $K/k$ be a Galois extension with group $G$, and suppose $S\subset G$ contains the identity. For any prime $\mathfrak{p}$ of $k$, let $δ_{\mathfrak{p}}$ be the proportion of the primes of $K$ above $\mathfrak{p}$ whose Frobenius lies in $S$. Then $\sum_{N\mathfrak{p}\le X}δ_{\mathfrak{p}}\,\log N\mathfrak{p}/N\mathfrak{p}\le(|S|/|G|+\varepsilon)\log X+O(1)$ holds uniformly in $K$ as soon as $X\ge d_K^{C}$, with $C$ and the implied constant depending only on $\varepsilon$, $C$, and the degree $[K:\mathbb{Q}]$. The proof is the argument of OpenAI in the form given by Kerr, simple vanishing at the points $(ιθ)_{ι\in S}$, with $θ$ in a box in $\mathcal{O}_K$, and Masser's zero estimate. The field enters only through Masser's invariant of the lattice of points, which we compute to be $[K:\mathbb{Q}]/|S|$ by averaging over the Galois group of the normal closure. Over $\mathbb{Q}$, the determinant method gives sharper Chebotarev bounds, and we explain the comparison.
Source: arXiv (Cornell University)
Author: Banks, William D.
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223358800