{"paper":{"title":"An analytic method for bounding $\\psi(x)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.NT","authors_text":"Jan B\\\"uthe","submitted_at":"2015-11-06T11:10:12Z","abstract_excerpt":"In this paper we present an analytic altorithm which calculates almost sharp bounds for the normalized error term $(t-\\psi(t))/\\sqrt{t}$ for $t\\leq x$ in expected run time $O(x^{1/2+\\varepsilon})$ for every $\\varepsilon>0$. The method has been implemented and used to calculate the bound $|\\psi(t) - t| \\leq 0.94 \\sqrt{t}$ for $11< t\\leq 10^{19}$. In particular, this bound implies that $\\operatorname{li}(t) - \\pi(t) > 0$ for $t\\in [2,10^{19}]$, which gives an improved lower bound for the Skewes number."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.02032","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"}