{"paper":{"title":"Extreme values of the Dedekind $\\Psi$ function","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":"2010-11-08T15:21:51Z","abstract_excerpt":"Let $\\Psi(n):=n\\prod_{p | n}(1+\\frac{1}{p})$ denote the Dedekind $\\Psi$ function. Define, for $n\\ge 3,$ the ratio $R(n):=\\frac{\\Psi(n)}{n\\log\\log n}.$ We prove unconditionally that $R(n)< e^\\gamma$ for $n\\ge 31.$ Let $N_n=2...p_n$ be the primorial of order $n.$ We prove that the statement $R(N_n)>\\frac{e^\\gamma}{\\zeta(2)}$ for $n\\ge 3$ is equivalent to the Riemann Hypothesis."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1825","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"}