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Define the ratio $R_t(n):=\\frac{\\Psi_t(n)}{n\\log\\log n}.$ W"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1012.0671","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-03T09:25:19Z","cross_cats_sorted":[],"title_canon_sha256":"2efac7e1cab3d3b853a09e2faf90bb7264ea4e75803a94b5c4e949ab447edb6a","abstract_canon_sha256":"19861e59fe7c32450b6a34b725666750dc2d7ec388fef081f3c8a58edd28c953"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:45.072744Z","signature_b64":"zHR3McU1s4VAEdNqJdzZLfADpXHFyTuLPlL3QtLPASWSoAmeUC5Yg7fS193lAcNIY58EXjp06teFTlE0lgsiBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"150186028681c32f8121d2bf82a773340b7d6887ae26c20464d622eb363b3879","last_reissued_at":"2026-05-18T04:06:45.072130Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:45.072130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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