{"paper":{"title":"Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dimitrios Chatzakos, Dmitry Frolenkov, Giacomo Cherubini, Niko Laaksonen, Olga Balkanova","submitted_at":"2017-12-04T02:28:13Z","abstract_excerpt":"For $\\Gamma$ a cofinite Kleinian group acting on $\\mathbb{H}^3$, we study the Prime Geodesic Theorem on $M=\\Gamma \\backslash \\mathbb{H}^3$, which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on $M$. Let $E_{\\Gamma}(X)$ be the error in the counting of prime geodesics with length at most $\\log X$. For the Picard manifold, $\\Gamma=\\mathrm{PSL}(2,\\mathbb{Z}[i])$, we improve the classical bound of Sarnak, $E_{\\Gamma}(X)=O(X^{5/3+\\epsilon})$, to $E_{\\Gamma}(X)=O(X^{13/8+\\epsilon})$. In the process we obtain a mean subconvexity estimate for the Rankin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.00880","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}