{
  "abstract": "We establish an asymptotic formula for the first moment of the central values of quadratic Dirichlet $L$-functions in the even hyperelliptic ensemble $\\mathcal{H}_{2g+2}$ over $\\mathbb{F}_q[x]$, for every fixed odd prime power $q$. Working with the completed $L$-function yields an exact two-term central-value formula with truncation levels $g$ and $g-1$; combining the two truncations before the square-dual contour shifts leaves the three cubic points $z^3 = q^{-4}$ as the only secondary poles, and they contribute the secondary term $q^{2g/3+2}(a_g g + b_g)$ with an error $O_\\varepsilon(q^{g(1+\\varepsilon)/2})$. The coefficients are real and $m \\mapsto (a_m, b_m)$ has minimal period exactly three, so the coefficient of $g$ at the order $q^{2g/3}$ depends on $g$ modulo 3, in contrast with the earlier even-degree formulas; the difference is confirmed numerically.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12764",
  "author_total": 1,
  "authors": [
    {
      "name": "Hwanyup Jung"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a86aht1xzrywa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12764",
  "fulltext_url": null,
  "id": "a86aht1xzrywa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12764",
  "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"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7212887111",
  "pdf_url": "https://arxiv.org/pdf/2609.12764",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a86aht1xzrywa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12764",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12764"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The first moment of quadratic Dirichlet $L$-functions in the even hyperelliptic ensemble",
  "title_text": "The first moment of quadratic Dirichlet L-functions in the even hyperelliptic ensemble",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/a86aht1xzrywa",
  "venue": "arXiv",
  "versions": []
}