{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:6XOGI7DPGYZWTEFPTOPHAMDRYG","short_pith_number":"pith:6XOGI7DP","schema_version":"1.0","canonical_sha256":"f5dc647c6f36336990af9b9e703071c19481a6c8b5e7a63001f49acb5d1ab54a","source":{"kind":"arxiv","id":"1011.4262","version":1},"attestation_state":"computed","paper":{"title":"The distribution functions of $\\sigma(n)/n$ and $n/\\phi(n)$, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andreas Weingartner","submitted_at":"2010-11-18T19:35:34Z","abstract_excerpt":"Let $\\sigma(n)$ be the sum of the positive divisors of $n$, and let $A(t)$ be the natural density of the set of positive integers $n$ satisfying $\\sigma(n)/n \\ge t$. We give an improved asymptotic result for $\\log A(t)$ as $t$ grows unbounded. The same result holds if $\\sigma(n)/n$ is replaced by $n/\\phi(n)$, where $\\phi(n)$ is Euler's totient function."},"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":"1011.4262","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-11-18T19:35:34Z","cross_cats_sorted":[],"title_canon_sha256":"b554e2e39d4e13ff1c1258c141f832cc15679acd926c99ff27d6f277dce1647e","abstract_canon_sha256":"09d3c295a46cb76060dfc83fd13d929099b4001537364371449c419c9f64b78f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:35:36.160041Z","signature_b64":"uZBPgi6Wo1h7ifXOA3nil1tSTc3QDe9fxErY3MJ0pimqSBc0j5oPpppJbsyEQYU/ehW7Qc23NMUIEEMQAG2hCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f5dc647c6f36336990af9b9e703071c19481a6c8b5e7a63001f49acb5d1ab54a","last_reissued_at":"2026-05-18T04:35:36.159422Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:35:36.159422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The distribution functions of $\\sigma(n)/n$ and $n/\\phi(n)$, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andreas Weingartner","submitted_at":"2010-11-18T19:35:34Z","abstract_excerpt":"Let $\\sigma(n)$ be the sum of the positive divisors of $n$, and let $A(t)$ be the natural density of the set of positive integers $n$ satisfying $\\sigma(n)/n \\ge t$. We give an improved asymptotic result for $\\log A(t)$ as $t$ grows unbounded. The same result holds if $\\sigma(n)/n$ is replaced by $n/\\phi(n)$, where $\\phi(n)$ is Euler's totient function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.4262","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1011.4262","created_at":"2026-05-18T04:35:36.159519+00:00"},{"alias_kind":"arxiv_version","alias_value":"1011.4262v1","created_at":"2026-05-18T04:35:36.159519+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.4262","created_at":"2026-05-18T04:35:36.159519+00:00"},{"alias_kind":"pith_short_12","alias_value":"6XOGI7DPGYZW","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_16","alias_value":"6XOGI7DPGYZWTEFP","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_8","alias_value":"6XOGI7DP","created_at":"2026-05-18T12:26:04.259169+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6XOGI7DPGYZWTEFPTOPHAMDRYG","json":"https://pith.science/pith/6XOGI7DPGYZWTEFPTOPHAMDRYG.json","graph_json":"https://pith.science/api/pith-number/6XOGI7DPGYZWTEFPTOPHAMDRYG/graph.json","events_json":"https://pith.science/api/pith-number/6XOGI7DPGYZWTEFPTOPHAMDRYG/events.json","paper":"https://pith.science/paper/6XOGI7DP"},"agent_actions":{"view_html":"https://pith.science/pith/6XOGI7DPGYZWTEFPTOPHAMDRYG","download_json":"https://pith.science/pith/6XOGI7DPGYZWTEFPTOPHAMDRYG.json","view_paper":"https://pith.science/paper/6XOGI7DP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1011.4262&json=true","fetch_graph":"https://pith.science/api/pith-number/6XOGI7DPGYZWTEFPTOPHAMDRYG/graph.json","fetch_events":"https://pith.science/api/pith-number/6XOGI7DPGYZWTEFPTOPHAMDRYG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6XOGI7DPGYZWTEFPTOPHAMDRYG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6XOGI7DPGYZWTEFPTOPHAMDRYG/action/storage_attestation","attest_author":"https://pith.science/pith/6XOGI7DPGYZWTEFPTOPHAMDRYG/action/author_attestation","sign_citation":"https://pith.science/pith/6XOGI7DPGYZWTEFPTOPHAMDRYG/action/citation_signature","submit_replication":"https://pith.science/pith/6XOGI7DPGYZWTEFPTOPHAMDRYG/action/replication_record"}},"created_at":"2026-05-18T04:35:36.159519+00:00","updated_at":"2026-05-18T04:35:36.159519+00:00"}