{
  "abstract": "We give an asymptotic formula for the number of Markov triples with largest entry below $x$. We show that the error term is $o(\\log x)$, which improves upon the previous best known results due to McShane and Rivin. We give an explicit error based on classical zero-free regions of the Riemann zeta function, and a sharper estimate under the quasi-Riemann hypothesis. The proof uses the correspondence between Markov triples and simple closed geodesics on the modular punctured torus and a careful analysis of the McShane-Rivin unit ball.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36069",
  "author_total": 1,
  "authors": [
    {
      "name": "Morten S. Risager"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5bh301689rfcb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36069",
  "fulltext_url": null,
  "id": "5bh301689rfcb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36069",
  "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.GT"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7224114165",
  "pdf_url": "https://arxiv.org/pdf/2610.36069",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5bh301689rfcb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36069",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36069"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Counting Markov triples and simple geodesics",
  "title_text": "Counting Markov triples and simple geodesics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/5bh301689rfcb",
  "venue": "arXiv",
  "versions": []
}