{"paper":{"title":"An asymptotic Robin inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Patrick Sol\\'e, Yuyang Zhu","submitted_at":"2015-12-31T09:48:20Z","abstract_excerpt":"The conjectured Robin inequality for an integer $n>7!$ is $\\sigma(n)<e^\\gamma n \\log \\log n,$ where $\\gamma$ denotes Euler constant, and $\\sigma(n)=\\sum_{d | n} d $. Robin proved that this conjecture is equivalent to Riemann hypothesis (RH). Writing $D(n)=e^\\gamma n \\log \\log n-\\sigma(n),$ and $d(n)=\\frac{D(n)}{n},$ we prove unconditionally that $\\liminf_{n \\rightarrow \\infty} d(n)=0.$ The main ingredients of the proof are an estimate for Chebyshev summatory function, and an effective version of Mertens third theorem due to Rosser and Schoenfeld. A new criterion for RH depending solely on $\\li"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.03384","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":""},"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"}