{"paper":{"title":"Robin inequality for $7-$free integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michel Planat (FEMTO-ST), Patrick Sol\\'e","submitted_at":"2010-12-03T09:25:19Z","abstract_excerpt":"Recall that an integer is $t-$free iff it is not divisible by $p^t$ for some prime $p.$ We give a method to check Robin inequality $\\sigma(n) < e^\\gamma n\\log\\log n,$ for $t-$free integers $n$ and apply it for $t=6,7.$ We introduce $\\Psi_t,$ a generalization of Dedekind $\\Psi$ function defined for any integer $t\\ge 2$ by $$\\Psi_t(n):=n\\prod_{p | n}(1+1/p+...+1/p^{t-1}).$$ If $n$ is $t-$free then the sum of divisor function $\\sigma(n)$ is $ \\le \\Psi_t(n).$ We characterize the champions for $x \\mapsto \\Psi_t(x)/x,$ as primorial numbers. Define the ratio $R_t(n):=\\frac{\\Psi_t(n)}{n\\log\\log n}.$ W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.0671","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"}