{
  "abstract": "e study large values of fixed derivatives of quadratic Dirichlet $L$-functions on the real segment $1/2\\leqσ\\leq1$. Assuming GRH, we prove lower bounds in terms of GCD sums and finite Euler products. At the central point, every fixed derivative order is compared with the same square-free GCD sum. In the mesoscopic band $σ=1/2+A/\\log_2X$, a renormalized resonator estimates contributions from both halves of the differentiated approximate functional equation. The resulting lower bound has a relative error. On a smaller boundary scale, a common finite product appears in both contributions to this lower bound. This allows us to locate the crossover between their derivative weights. The coarser exponential bounds have constant $1$ at the center and $\\mathrm e^{-A}$ in the mesoscopic band. For fixed $1/2<σ<1$, the lower bound includes the full typical logarithmic weight of the resonating Euler product. Burgess truncation improves the exponential constant when a shorter cutoff allows a longer resonator. Near the $1$-line, our lower bounds involve Dickman constants on the $1/\\log_2X$ scale. We also prove a uniform transition estimate outside a shrinking neighborhood of the central point. All results hold in the positive even subfamily of fundamental discriminants.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24958",
  "author_total": 5,
  "authors": [
    {
      "name": "Zikang Dong"
    },
    {
      "name": "Ruihua Wang"
    },
    {
      "name": "Weijia Wang"
    },
    {
      "name": "Hao Zhang"
    },
    {
      "name": "Shengbo Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/20mb5fcbrjavx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24958",
  "fulltext_url": null,
  "id": "20mb5fcbrjavx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24958",
  "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": "W7223056487",
  "pdf_url": "https://arxiv.org/pdf/2610.24958",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/20mb5fcbrjavx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24958",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24958"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Large values of derivatives of quadratic Dirichlet $L$-functions",
  "title_text": "Large values of derivatives of quadratic Dirichlet L-functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/20mb5fcbrjavx",
  "venue": "arXiv",
  "versions": []
}