Landau-Siegel zeros over a base field and effective Brauer-Siegel
Abstract
Let k be a number field. We prove that for every quadratic extension F/k, every real zero β∈(0,1) of L(s,χF/k) satisfies (1−β)log dF≥ 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 hK RK ≥ c(k) dK1/2/((2π)[K:ℚ]log dK). 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⊂ G contains the identity. For any prime p of k, let δp be the proportion of the primes of K above p whose Frobenius lies in S. Then ∑Np≤ Xδp log Np/Np≤(|S|/|G|+ε)log X+O(1) holds uniformly in K as soon as X≥ dKC, with C and the implied constant depending only on ε, C, and the degree [K:ℚ]. The proof is the argument of OpenAI in the form given by Kerr, simple vanishing at the points (ιθ)ι∈ S, with θ in a box in 𝒪K, and Masser's zero estimate. The field enters only through Masser's invariant of the lattice of points, which we compute to be [K:ℚ]/|S| by averaging over the Galois group of the normal closure. Over ℚ, the determinant method gives sharper Chebotarev bounds, and we explain the comparison.
Related
BibTeX
@misc{banks2026landausiegel,
title = {{Landau-Siegel zeros over a base field and effective Brauer-Siegel}},
author = {William D. Banks},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31402},
eprint = {2610.31402},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31402},
}
RIS
TY - UNPB
TI - Landau-Siegel zeros over a base field and effective Brauer-Siegel
AU - William D. Banks
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.31402
UR - https://arxiv.org/abs/2610.31402
AB - Let k be a number field. We prove that for every quadratic extension F/k, every real zero β∈(0,1) of L(s,χ_F/k) satisfies (1−β)log d_F≥ 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 ≥ c(k) d_K^1/2/((2π)^[K: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⊂ G contains the identity. For any prime p of k, let δ_p be the proportion of the primes of K above p whose Frobenius lies in S. Then ∑_Np≤ Xδ_p log Np/Np≤(|S|/|G|+ε)log X+O(1) holds uniformly in K as soon as X≥ d_K^C, with C and the implied constant depending only on ε, C, and the degree [K:ℚ]. The proof is the argument of OpenAI in the form given by Kerr, simple vanishing at the points (ιθ)_ι∈ S, with θ in a box in 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:ℚ]/|S| by averaging over the Galois group of the normal closure. Over ℚ, the determinant method gives sharper Chebotarev bounds, and we explain the comparison.
ER -