{
  "abstract": "We count the integral solutions of $x_1^2+x_2^2-x_3^2=D$ weighted by $Φ(x_3/M)$, for $M$ smaller than $\\sqrt D$. Such solutions correspond to binary quadratic forms of discriminant $D$, and the count becomes a sum, over the geodesics of discriminant $D$, of a function of their distance to the point $i$. We evaluate it by the spectral theory of automorphic forms. Throughout, $θ=7/64$. For square $D=n^2$ the geodesics run from cusp to cusp, and Hecke theory reduces them to the single geodesic from $0$ to $\\infty$. For every $n$ and both parities of $x_3$, we obtain an asymptotic formula with error $O(n^{1+θ+\\varepsilon}M^{-1/2})$, using the Kim-Sarnak bound and the large sieve of Deshouillers and Iwaniec (and the theta group for the parity of $x_3$). It is non-trivial for $M\\ge n^{71/96+\\varepsilon}$. This improves the range $n^{5/6+\\varepsilon}$ of the elementary method, and confirms the corrected main term of the author's earlier work. For non-square $D=f^2D_0$ the geodesics are closed. A Hecke relation for their traces reduces everything to the fundamental discriminant $D_0$. We obtain an error $O(D^{\\varepsilon}f^{1+θ}D_0^{2/3}M^{-1/2})$, using the formulas of Duke-Imamoglu-Toth and Baruch-Mao together with the cubic moment bound of Young and of Andersen-Wu. For $D\\equiv 5 \\bmod 8$ this counts all solutions, and the range $M\\ge D^{1/2-1/1332}$ of Friedlander and Iwaniec improves to $M\\ge D^{4/9+\\varepsilon}$. This paper was written in collaboration with Claude (Opus 5.5, High), an AI model developed by Anthropic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15706",
  "author_total": 1,
  "authors": [
    {
      "name": "Stephan Baier"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5c2zfavtk8btq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15706",
  "fulltext_url": null,
  "id": "5c2zfavtk8btq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15706",
  "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": "W7221780019",
  "pdf_url": "https://arxiv.org/pdf/2610.15706",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5c2zfavtk8btq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.15706",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15706"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Solutions of $x_1^2+x_2^2-x_3^2=D$ with small $x_3$: a spectral approach",
  "title_text": "Solutions of x_1^2+x_2^2−x_3^2=D with small x_3: a spectral approach",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/5c2zfavtk8btq",
  "venue": "arXiv",
  "versions": []
}