{"paper":{"title":"The n-th prime exponentially","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Matt Visser (Victoria University of Wellington)","submitted_at":"2025-04-20T02:35:53Z","abstract_excerpt":"From known effective bounds on the prime counting function of the form \\[ |\\pi(x)-\\mathrm{Li}(x)| < a \\;x \\;(\\ln x)^{b} \\; \\exp\\left(-{c}\\; \\sqrt{\\ln x}\\right); \\qquad (x \\geq x_0); \\] it is possible to establish exponentially tight effective upper and lower bounds on the prime number theorem: For $x \\geq x_*$ where $x_*\\leq \\max\\{x_0,17\\}$ we have: \\[ {\\mathrm{Li} \\over 1+a\\; (\\ln x)^{b+1} \\; \\exp\\left(-c\\; \\sqrt{\\ln x}\\right)} < \\pi(x) < {\\mathrm{Li} \\over 1-a \\;(\\ln x)^{b+1} \\; \\exp\\left(-c\\; \\sqrt{\\ln x}\\right)}. \\] Furthermore, it is possible to establish exponentially tight effective upp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.14458","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/2504.14458/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"}