{
  "abstract": "In [Liu26a], varying the length and moment parameters in the cubic-theta argument of [Ope26a] gave the zero-free half-plane $Re(s)>B_{\\mathrm{new}}=0.874957019420098946\\ldots$ for finite-order Hecke $L$-functions over $\\mathbb Q(\\sqrt{-3})$ and for Dirichlet $L$-functions, and its optimality in the specified exponent comparison system. In this paper, we extend the argument to finite-order Hecke $L$-functions over cyclotomic fields containing $μ_{12}$ and, by class-field descent, to every pair $(K,η)$ whose cyclic class field has metabelian normal closure over $\\mathbb Q$, including every finite-order character over every abelian number field. Then, adapting the averaged numerator estimate developed by Xu and Sang [XS26] for Hecke $L$-functions over $\\mathbb Q(\\sqrt{-3})$ and combining it with a limiting contradiction argument, we obtain the boundary $B_\\flat=0.874886647039964\\ldots$, the root near $0.8749$ of $133488B^4-376596B^3+384534B^2-167541B+26231$; it is the exact infimum of the refined comparison system, unattained by finite chains starting above it. Since the analytic argument uses only $μ_{12}\\subset F$, we run it over $F=K(ζ_{12})$ for an arbitrary number field $K$ and descend: every finite-order Hecke $L$-function over every number field is zero-free for $Re(s)>B_\\flat$, giving a uniform Quasi-Riemann Hypothesis over number fields. The analytic argument uses the exceptional-theta constructions and local calculations of the preprint [Ope26b].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26835",
  "author_total": 1,
  "authors": [
    {
      "name": "Baiying Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bvswqmwsd0y0v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26835",
  "fulltext_url": null,
  "id": "bvswqmwsd0y0v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26835",
  "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"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223017165",
  "pdf_url": "https://arxiv.org/pdf/2610.26835",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bvswqmwsd0y0v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:35Z",
      "native_id": "2610.26835",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26835"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Zero-free half-planes for Hecke $L$-functions over number fields and Quasi-Riemann Hypothesis",
  "title_text": "Zero-free half-planes for Hecke L-functions over number fields and Quasi-Riemann Hypothesis",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:53Z",
  "url": "https://research.draconitic.com/paper/bvswqmwsd0y0v",
  "venue": "arXiv",
  "versions": []
}