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Counting Markov triples and simple geodesics
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.
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BibTeX
@misc{risager2026counting,
title = {{Counting Markov triples and simple geodesics}},
author = {Morten S. Risager},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36069},
eprint = {2610.36069},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36069},
}
RIS
TY - UNPB
TI - Counting Markov triples and simple geodesics
AU - Morten S. Risager
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36069
UR - https://arxiv.org/abs/2610.36069
AB - 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.
ER -