Counting multigeodesics and the Selberg-Delange method

Abstract

We study the counting function 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 𝒫^m(X,x). We show that for fixed m as x→∞ the asymptotic is independent of the surface, whereas for 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 ω₁, defined on multigeodesics Γ as ω₁(Γ)=k−1 for Γ=(γ₁,...,γ_k) a k-multigeodesic. We show that as x→∞, the distribution of ω₁ in the set of multigeodesics of length ≤ 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 ω₁ which we relate to the Selberg zeta function.

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BibTeX RIS

BibTeX

@misc{karvonen2026counting,
  title = {{Counting multigeodesics and the Selberg-Delange method}},
  author = {Aapo Karvonen},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08165},
  eprint = {2611.08165},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08165},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Counting multigeodesics and the Selberg-Delange method
AU  - Aapo Karvonen
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08165
UR  - https://arxiv.org/abs/2611.08165
AB  - We study the counting function 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 𝒫^m(X,x). We show that for fixed m as x→∞ the asymptotic is independent of the surface, whereas for 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→∞, the distribution of ω_1 in the set of multigeodesics of length ≤ 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.
ER  -

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