{"paper":{"title":"New bounds in R.S. Lehman's estimates for the difference $\\pi\\left( x\\right) -li\\left( x\\right) $","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michael Revers","submitted_at":"2025-01-08T13:14:54Z","abstract_excerpt":"We denote by $\\pi\\left( x\\right) $ the usual prime counting function and let $li\\left( x\\right) $ the logarithmic integral of $x$. In 1966, R.S. Lehman came up with a new approach and an effective method for finding an upper bound where it is assured that a sign change occurs for $\\pi\\left( x\\right) -li\\left( x\\right) $ for some value $x$ not higher than this given bound. In this paper we provide further improvements on the error terms including an improvement upon Lehman's famous error term $S_{3}$ in his original paper. We are now able to eliminate the lower condition for the size-length $\\e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.04488","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/2501.04488/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"}