{
  "abstract": "We study the counting function $\\mathcal{P}^m(X,x)$ for $m$-multigeodesics of length at most $x$ on a compact hyperbolic surface $X$ where $m$ is a positive integer. Building on the Selberg-Delange method, originally developed for counting integers with exactly $m$ prime factors in the interval $[1,x]$, we prove a large $x$ asymptotic for $\\mathcal{P}^m(X,x)$. We show that for fixed $m$ as $x\\to\\infty$ the asymptotic is independent of the surface, whereas for $\\frac{m}{\\log x}$ bounded away from zero, the asymptotic depends on an infinite Euler product over the lengths of primitive closed geodesics. We also study the distribution of the function $ω_1$, defined on multigeodesics $Γ$ as $ω_1(Γ)=k-1$ for $Γ=(γ_1,...,γ_k)$ a $k$-multigeodesic. We show that as $x\\to\\infty$, the distribution of $ω_1$ in the set of multigeodesics of length $\\leq x$ is close to that of a Poisson random variable with parameter $\\log x$. Both of these theorems rely on studying the generating function of the random variable $ω_1$ which we relate to the Selberg zeta function.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08165",
  "author_total": 1,
  "authors": [
    {
      "name": "Aapo Karvonen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/19w3c72xsqbt4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08165",
  "fulltext_url": null,
  "id": "19w3c72xsqbt4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08165",
  "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.GT"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.SP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08165",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/19w3c72xsqbt4.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08165",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08165"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Counting multigeodesics and the Selberg-Delange method",
  "title_text": "Counting multigeodesics and the Selberg-Delange method",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/19w3c72xsqbt4",
  "venue": "arXiv",
  "versions": []
}