{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31402",
  "author_total": 1,
  "authors": [
    {
      "name": "William D. Banks"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e036f7kgttvhg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31402",
  "fulltext_url": null,
  "id": "e036f7kgttvhg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31402",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223358800",
  "pdf_url": "https://arxiv.org/pdf/2610.31402",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e036f7kgttvhg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.31402",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31402"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Landau-Siegel zeros over a base field and effective Brauer-Siegel",
  "title_text": "Landau-Siegel zeros over a base field and effective Brauer-Siegel",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/e036f7kgttvhg",
  "venue": "arXiv",
  "versions": []
}