Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Risager, Morten S.
Institution: University of Copenhagen
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224114165