{"paper":{"title":"Closed geodesics in short intervals for random hyperbolic surfaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.GT","authors_text":"Zeev Rudnick","submitted_at":"2026-05-21T06:47:53Z","abstract_excerpt":"We study the distribution of closed geodesics in short intervals on random hyperbolic surfaces of large genus, and compare it with the classical problem of primes in short intervals. Viewing the surface $M$ as a random point in moduli space equipped with the Weil--Petersson measure, we investigate the random variable $\\Psi_M(x;H)$ counting closed geodesics with norms in the interval $[X, X+H]$, weighted by primitive length, where $H=o(X)$. This is analogous to the Chebyshev function in prime number theory.\n  Our main result establishes that in the large genus limit, \\[ \\lim_{g\\to \\infty}\\mathr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22059","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.22059/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}