{"paper":{"title":"Bykovskii-type theorem for the Picard manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andr\\'as Bir\\'o, Antal Balog, Giacomo Cherubini, Niko Laaksonen","submitted_at":"2019-11-05T14:33:06Z","abstract_excerpt":"We generalise a result of Bykovskii to the Gaussian integers and prove an asymptotic formula for the prime geodesic theorem in short intervals on the Picard manifold. Previous works show that individually the remainder is bounded by $O(X^{13/8+\\epsilon})$ and $O(X^{3/2+\\theta+\\epsilon})$, where $\\theta$ is the subconvexity exponent for quadratic Dirichlet $L$-functions over $\\mathbb{Q}(i)$. By combining arithmetic methods with estimates for a spectral exponential sum and a smooth explicit formula, we obtain an improvement for both of these exponents. Moreover, by assuming two standard conjectu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.01800","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1911.01800/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"}