{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:D4MVO3OWECPB2IZYOXBPZYDI4G","short_pith_number":"pith:D4MVO3OW","schema_version":"1.0","canonical_sha256":"1f19576dd6209e1d233875c2fce068e196b7d8979fc01d96df4a76dab58de96f","source":{"kind":"arxiv","id":"1104.3264","version":2},"attestation_state":"computed","paper":{"title":"The distribution of totients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kevin Ford","submitted_at":"2011-04-16T21:43:56Z","abstract_excerpt":"New version of my 1998 article. The method of proof of the main results follows the original, but there are many simplifications/streamlining of arguments, especially Lemma 3.6 (new Lemma 3.7). Fixed small error in proof of lower bound for V_k(x) (see the paragraph after (5.20)), fixed the statement and proof of Theorem 3. New, precise way to relate sums to volumes (Lemmas 3.1, 3.9) and provided full details in Section 6 of the proofs of Theorems 10-13. Slightly different versions of Theorems 10,11,12,14 (qualitatively the same) with simpler proofs. Sections 7 and 8 combined (new section 7). A"},"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":"1104.3264","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-04-16T21:43:56Z","cross_cats_sorted":[],"title_canon_sha256":"892679123968442533a97a1a81d0f6ba8d94d4ca3966d537f79b772fea226a51","abstract_canon_sha256":"ee49a3c81f5fe53c3f0fd0e69e3a8e64f55af771e8c384a870de0eabc6d3cd9c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:32.125513Z","signature_b64":"Vng5zwHUDcztfMzApw3ava1GtO/GEWcU7fggw9/JQ4tdA4dIHoj+7tq3u8owuBEcJfmtAw68WYuYZ9yU24QxBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1f19576dd6209e1d233875c2fce068e196b7d8979fc01d96df4a76dab58de96f","last_reissued_at":"2026-05-18T03:18:32.124953Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:32.124953Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The distribution of totients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kevin Ford","submitted_at":"2011-04-16T21:43:56Z","abstract_excerpt":"New version of my 1998 article. The method of proof of the main results follows the original, but there are many simplifications/streamlining of arguments, especially Lemma 3.6 (new Lemma 3.7). Fixed small error in proof of lower bound for V_k(x) (see the paragraph after (5.20)), fixed the statement and proof of Theorem 3. New, precise way to relate sums to volumes (Lemmas 3.1, 3.9) and provided full details in Section 6 of the proofs of Theorems 10-13. Slightly different versions of Theorems 10,11,12,14 (qualitatively the same) with simpler proofs. Sections 7 and 8 combined (new section 7). A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.3264","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1104.3264","created_at":"2026-05-18T03:18:32.125028+00:00"},{"alias_kind":"arxiv_version","alias_value":"1104.3264v2","created_at":"2026-05-18T03:18:32.125028+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.3264","created_at":"2026-05-18T03:18:32.125028+00:00"},{"alias_kind":"pith_short_12","alias_value":"D4MVO3OWECPB","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"D4MVO3OWECPB2IZY","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"D4MVO3OW","created_at":"2026-05-18T12:26:26.731475+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/D4MVO3OWECPB2IZYOXBPZYDI4G","json":"https://pith.science/pith/D4MVO3OWECPB2IZYOXBPZYDI4G.json","graph_json":"https://pith.science/api/pith-number/D4MVO3OWECPB2IZYOXBPZYDI4G/graph.json","events_json":"https://pith.science/api/pith-number/D4MVO3OWECPB2IZYOXBPZYDI4G/events.json","paper":"https://pith.science/paper/D4MVO3OW"},"agent_actions":{"view_html":"https://pith.science/pith/D4MVO3OWECPB2IZYOXBPZYDI4G","download_json":"https://pith.science/pith/D4MVO3OWECPB2IZYOXBPZYDI4G.json","view_paper":"https://pith.science/paper/D4MVO3OW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1104.3264&json=true","fetch_graph":"https://pith.science/api/pith-number/D4MVO3OWECPB2IZYOXBPZYDI4G/graph.json","fetch_events":"https://pith.science/api/pith-number/D4MVO3OWECPB2IZYOXBPZYDI4G/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D4MVO3OWECPB2IZYOXBPZYDI4G/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D4MVO3OWECPB2IZYOXBPZYDI4G/action/storage_attestation","attest_author":"https://pith.science/pith/D4MVO3OWECPB2IZYOXBPZYDI4G/action/author_attestation","sign_citation":"https://pith.science/pith/D4MVO3OWECPB2IZYOXBPZYDI4G/action/citation_signature","submit_replication":"https://pith.science/pith/D4MVO3OWECPB2IZYOXBPZYDI4G/action/replication_record"}},"created_at":"2026-05-18T03:18:32.125028+00:00","updated_at":"2026-05-18T03:18:32.125028+00:00"}