{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:JYP5WUQQUOOYMNRRWJ2CQNNZDM","short_pith_number":"pith:JYP5WUQQ","schema_version":"1.0","canonical_sha256":"4e1fdb5210a39d863631b2742835b91b15c569a69e96ef6f50f0f48c57cf722f","source":{"kind":"arxiv","id":"math/0607473","version":5},"attestation_state":"computed","paper":{"title":"Integers with a divisor in (y,2y]","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kevin Ford","submitted_at":"2006-07-19T17:18:45Z","abstract_excerpt":"We give a relatively short proof of one of the central cases of the main theorem from the paper \"The distribution of integers with a divisor in a given interval\", math.NT/0401223. Namely, we determine the order of magnitude of the number of integers <=x with a divisor in (y,2y]. The lower bound uses a different argument than that in the aforementioned paper. As a corollary, we deduce the order of magnitude for the number of distinct products in an N x N multiplication table."},"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":"math/0607473","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2006-07-19T17:18:45Z","cross_cats_sorted":[],"title_canon_sha256":"1d02af2f3261e1d47f5280cf8a800035ca75729421a3723e2d1139b1465abf73","abstract_canon_sha256":"17ded316a042f2ae511daca862baaaa462a6fd24343efa5de2ce2c61df81c55b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:38.567711Z","signature_b64":"Qjo9cy/QSidtwRD8m9FKttBX3xJv7h4sxiymYav5YThQ4eEgl63FYzS/VwPVTHdrl0+QcPaArCEGepFc6WqQDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e1fdb5210a39d863631b2742835b91b15c569a69e96ef6f50f0f48c57cf722f","last_reissued_at":"2026-05-18T03:30:38.566839Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:38.566839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Integers with a divisor in (y,2y]","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kevin Ford","submitted_at":"2006-07-19T17:18:45Z","abstract_excerpt":"We give a relatively short proof of one of the central cases of the main theorem from the paper \"The distribution of integers with a divisor in a given interval\", math.NT/0401223. Namely, we determine the order of magnitude of the number of integers <=x with a divisor in (y,2y]. The lower bound uses a different argument than that in the aforementioned paper. As a corollary, we deduce the order of magnitude for the number of distinct products in an N x N multiplication table."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607473","kind":"arxiv","version":5},"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":"math/0607473","created_at":"2026-05-18T03:30:38.567002+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0607473v5","created_at":"2026-05-18T03:30:38.567002+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0607473","created_at":"2026-05-18T03:30:38.567002+00:00"},{"alias_kind":"pith_short_12","alias_value":"JYP5WUQQUOOY","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_16","alias_value":"JYP5WUQQUOOYMNRR","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_8","alias_value":"JYP5WUQQ","created_at":"2026-05-18T12:25:54.717736+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/JYP5WUQQUOOYMNRRWJ2CQNNZDM","json":"https://pith.science/pith/JYP5WUQQUOOYMNRRWJ2CQNNZDM.json","graph_json":"https://pith.science/api/pith-number/JYP5WUQQUOOYMNRRWJ2CQNNZDM/graph.json","events_json":"https://pith.science/api/pith-number/JYP5WUQQUOOYMNRRWJ2CQNNZDM/events.json","paper":"https://pith.science/paper/JYP5WUQQ"},"agent_actions":{"view_html":"https://pith.science/pith/JYP5WUQQUOOYMNRRWJ2CQNNZDM","download_json":"https://pith.science/pith/JYP5WUQQUOOYMNRRWJ2CQNNZDM.json","view_paper":"https://pith.science/paper/JYP5WUQQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0607473&json=true","fetch_graph":"https://pith.science/api/pith-number/JYP5WUQQUOOYMNRRWJ2CQNNZDM/graph.json","fetch_events":"https://pith.science/api/pith-number/JYP5WUQQUOOYMNRRWJ2CQNNZDM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JYP5WUQQUOOYMNRRWJ2CQNNZDM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JYP5WUQQUOOYMNRRWJ2CQNNZDM/action/storage_attestation","attest_author":"https://pith.science/pith/JYP5WUQQUOOYMNRRWJ2CQNNZDM/action/author_attestation","sign_citation":"https://pith.science/pith/JYP5WUQQUOOYMNRRWJ2CQNNZDM/action/citation_signature","submit_replication":"https://pith.science/pith/JYP5WUQQUOOYMNRRWJ2CQNNZDM/action/replication_record"}},"created_at":"2026-05-18T03:30:38.567002+00:00","updated_at":"2026-05-18T03:30:38.567002+00:00"}