{
  "abstract": "We study the $β=2$ partition function $\\int_{|h| \\leq \\log ^θ q/2}|L(1/2+ih,χ)|^2dh$ for typical Dirichlet characters modulo a large prime $q$ and $θ\\in (-1/3,0]$ motivated by a $q$-analogue of the Saksman--Webb conjectures. When $θ<0$, we use an extension of Harper's randomisation argument with different localisations to recover moment upper bounds consistent with the critical normalisation predicted there. As an application, we prove that for $q(1-o(1))$ Dirichlet characters modulo $q$ with $U(q) \\to \\infty$, $$\\max_{|h| \\leq 1/2}|L(1/2+ih,χ)|\\ll \\frac{e^{U(q)}\\log q\\log_3 q(\\log_4 q)^{1/2}}{(\\log_2 q)^{3/4}},$$ establishing an upper bound matching the predictions of the $q$-analogue of the Fyodorov--Hiary--Keating conjectures up to second order. Our proofs adapt Harper's argument in \\cite{harper2019partition}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.09906",
  "author_total": 1,
  "authors": [
    {
      "name": "Christopher Atherfold"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ag65ajc14vf0w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.09906",
  "fulltext_url": null,
  "id": "ag65ajc14vf0w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.09906",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7202167482",
  "pdf_url": "https://arxiv.org/pdf/2608.09906",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ag65ajc14vf0w.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:28Z",
      "native_id": "2608.09906",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.09906"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The $β=2$ partition function of Dirichlet $L$-functions and typical maxima",
  "title_text": "The β=2 partition function of Dirichlet L-functions and typical maxima",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/ag65ajc14vf0w",
  "venue": "arXiv",
  "versions": []
}