{
  "abstract": "We prove that the sum of class numbers of primitive indefinite binary quadratic forms with integer coefficients over positive nonsquare discriminants less than $X$ is $O(X^{3/2}\\sqrt{\\log\\log X}/(\\log X)^{3/2})$. This improves the longstanding classical upper bound $O(X^{3/2}/\\log X)$ and implies the same upper bound for the corresponding sum over real quadratic fields. Our main input is a uniform upper bound for the number of small fundamental solutions to Pell's equation. In particular, we prove that, for almost all nonsquare positive integers $D$, the fundamental solution $\\varepsilon_D$ of $t^2-Du^2=1$ is larger than any fixed power of $D$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28001",
  "author_total": 1,
  "authors": [
    {
      "name": "Yijie Diao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/atn3twzsayf30.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28001",
  "fulltext_url": null,
  "id": "atn3twzsayf30",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28001",
  "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": "W7223153041",
  "pdf_url": "https://arxiv.org/pdf/2610.28001",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/atn3twzsayf30.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28001",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28001"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the average of class numbers of indefinite binary quadratic forms",
  "title_text": "On the average of class numbers of indefinite binary quadratic forms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/atn3twzsayf30",
  "venue": "arXiv",
  "versions": []
}