{
  "abstract": "Let $q$ tend to infinity through odd primes, let $T=T(q)$, and put $\\ell=\\log(qT/2π)$. Suppose that $\\ell^s=o(T)$ for some fixed $s>1$ and that \\[ λ_{\\rm bw}:=\\min\\left\\{1, \\liminf_{\\substack{q\\to\\infty\\\\q\\ \\mathrm{prime}}} \\frac{\\log(q-1)}{\\ell}\\right\\}>0. \\] For the unweighted family of the $q-2$ nonprincipal characters modulo $q$, we prove unconditional lower bounds, relative to the total zero multiplicity in $(T,2T]$, of $2-1/c_{λ_{\\rm bw}}^*$ for both simple and distinct critical-line zeros and of $\\tfrac12(3-1/c_{λ_{\\rm bw}}^*)$ for all distinct zeros, where \\[ c_λ^*=\\frac{\\sqrt2\\tan(λ/\\sqrt2)} {1+(λ/\\sqrt2)\\tan(λ/\\sqrt2)}. \\] When $\\log T=o(\\log q)$, this gives respectively $0.6725007036\\ldots$ and $0.8362503518\\ldots$; in particular it covers every fixed $T=(\\log q)^A$ with $A>1$. As the growing-height companion to the authors' public mesoscopic shrinking-height theorem, this article is organized around the different localization technology required when $T$ grows. An abstract theorem converts complex-strip Gevrey decay, sampling density, and a local multiset count into stretched-exponential localization in trace and nuclear norm. In the Dirichlet specialization the strip growth is $X^{1/4}$, where $X=\\exp(λ\\ell)$, and a buffer $D_0=(K\\ell)^s$ absorbs the remote-zero contribution uniformly in the character. Quantitative finite-sampling end-effect bounds, a finite explicit-formula matrix, character-averaged first and second traces, hyperbolic functional-equation blocks, and a rank--trace inequality then yield the three zero statistics. The shared finite-inertia mechanism and the Montgomery--Taylor constant are not claimed as new. No form of GRH is assumed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27808",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhixu Hua"
    },
    {
      "name": "Xiufan Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/etj7xe9wk6h7g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27808",
  "fulltext_url": null,
  "id": "etj7xe9wk6h7g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27808",
  "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": "W7214181669",
  "pdf_url": "https://arxiv.org/pdf/2609.27808",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/etj7xe9wk6h7g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27808",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27808"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Gevrey localization for simple and distinct zeros in a prime-modulus Dirichlet family",
  "title_text": "Gevrey localization for simple and distinct zeros in a prime-modulus Dirichlet family",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/etj7xe9wk6h7g",
  "venue": "arXiv",
  "versions": []
}