{"paper":{"title":"Sharper bounds for the error term in the Prime Number Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrew Fiori, Habiba Kadiri, Joshua Swidinsky","submitted_at":"2022-06-25T05:05:10Z","abstract_excerpt":"We provide very effective methods to convert both asymptotic and explicit numeric bounds on the prime counting function $\\psi(x)$ to bounds of the same type on both $\\theta(x)$ and $\\pi(x)$. This follows up our previous work on $\\psi(x)$ in \\cite{FKS}, and prove that $ | \\pi(x) - \\mathrm{Li}(x) | \\leq 9.2211\\, x\\sqrt{\\log(x)} \\exp \\big( -0.8476 \\sqrt{\\log(x)} \\big) $ for all $x\\ge 2$. Additionally, we are able to obtain the best numeric bounds for $x$ on a very large interval (all $x$ up to $\\exp(1.8\\cdot10^9)$)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.12557","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/2206.12557/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"}