{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:JTCFE534LQKRZAA7OSIL7ALFEQ","short_pith_number":"pith:JTCFE534","canonical_record":{"source":{"id":"1103.2106","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-03-10T19:03:52Z","cross_cats_sorted":[],"title_canon_sha256":"8aa6de77ab6a7a8a07f8c684c529391a5531c47563ba1edac562def8836f97a9","abstract_canon_sha256":"fd9346d1cc613d357523d997d89c7822aee1e7303d904b38bac4bd87cfd49019"},"schema_version":"1.0"},"canonical_sha256":"4cc452777c5c151c801f7490bf8165242fd55c182e6d760da475c1f214611bd4","source":{"kind":"arxiv","id":"1103.2106","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.2106","created_at":"2026-05-18T04:27:02Z"},{"alias_kind":"arxiv_version","alias_value":"1103.2106v1","created_at":"2026-05-18T04:27:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.2106","created_at":"2026-05-18T04:27:02Z"},{"alias_kind":"pith_short_12","alias_value":"JTCFE534LQKR","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JTCFE534LQKRZAA7","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JTCFE534","created_at":"2026-05-18T12:26:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:JTCFE534LQKRZAA7OSIL7ALFEQ","target":"record","payload":{"canonical_record":{"source":{"id":"1103.2106","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-03-10T19:03:52Z","cross_cats_sorted":[],"title_canon_sha256":"8aa6de77ab6a7a8a07f8c684c529391a5531c47563ba1edac562def8836f97a9","abstract_canon_sha256":"fd9346d1cc613d357523d997d89c7822aee1e7303d904b38bac4bd87cfd49019"},"schema_version":"1.0"},"canonical_sha256":"4cc452777c5c151c801f7490bf8165242fd55c182e6d760da475c1f214611bd4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:27:02.964232Z","signature_b64":"C+0quGD/A+6ET0hV6A2zDtXbK6rCK83+JI83DVcdFwLnWJc/sqPn0+s+k/R90G7Hvcn428+NTmbC5xqZaf7hCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4cc452777c5c151c801f7490bf8165242fd55c182e6d760da475c1f214611bd4","last_reissued_at":"2026-05-18T04:27:02.963557Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:27:02.963557Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.2106","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:27:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sKNI70zgWY9X7dolcpmIWSQyAlX3RbXzTw53bH1ZtKXwKqk7pK5uDqWI+xof4NmYWIssNr8pvdMwIth7NkdFBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T20:46:31.122711Z"},"content_sha256":"2a7215c86224b2352f4caaeef29e5900b09f64bea2f5cc257e63d3190aab8f4b","schema_version":"1.0","event_id":"sha256:2a7215c86224b2352f4caaeef29e5900b09f64bea2f5cc257e63d3190aab8f4b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:JTCFE534LQKRZAA7OSIL7ALFEQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a paper of K. Soundararajan on smooth numbers in arithmetic progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Adam J. Harper","submitted_at":"2011-03-10T19:03:52Z","abstract_excerpt":"In a recent paper, K. Soundararajan showed, roughly speaking, that the integers smaller than x whose prime factors are less than y are asymptotically equidistributed in arithmetic progressions to modulus q, provided that y^{4\\sqrt{e}-\\delta} \\geq q and that y is neither too large nor too small compared with x. We show that these latter restrictions on y are unnecessary, thereby proving a conjecture of Soundararajan. Our argument uses a simple majorant principle for trigonometric sums to handle a saddle point that is close to 1."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.2106","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:27:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JYDMKQ6PZjLRA1JDe2HszuI5pRQhv2sLX2F2sJHKgiRx7MhuwzlNu2mgqY1kETTRtNA9yJKXX0jKVCumnlCCCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T20:46:31.123515Z"},"content_sha256":"64dcc124df7dc841557416e964448d8c2aaddf828e245cf6c43c860015e0e7fb","schema_version":"1.0","event_id":"sha256:64dcc124df7dc841557416e964448d8c2aaddf828e245cf6c43c860015e0e7fb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JTCFE534LQKRZAA7OSIL7ALFEQ/bundle.json","state_url":"https://pith.science/pith/JTCFE534LQKRZAA7OSIL7ALFEQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JTCFE534LQKRZAA7OSIL7ALFEQ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-31T20:46:31Z","links":{"resolver":"https://pith.science/pith/JTCFE534LQKRZAA7OSIL7ALFEQ","bundle":"https://pith.science/pith/JTCFE534LQKRZAA7OSIL7ALFEQ/bundle.json","state":"https://pith.science/pith/JTCFE534LQKRZAA7OSIL7ALFEQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JTCFE534LQKRZAA7OSIL7ALFEQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:JTCFE534LQKRZAA7OSIL7ALFEQ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"fd9346d1cc613d357523d997d89c7822aee1e7303d904b38bac4bd87cfd49019","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-03-10T19:03:52Z","title_canon_sha256":"8aa6de77ab6a7a8a07f8c684c529391a5531c47563ba1edac562def8836f97a9"},"schema_version":"1.0","source":{"id":"1103.2106","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.2106","created_at":"2026-05-18T04:27:02Z"},{"alias_kind":"arxiv_version","alias_value":"1103.2106v1","created_at":"2026-05-18T04:27:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.2106","created_at":"2026-05-18T04:27:02Z"},{"alias_kind":"pith_short_12","alias_value":"JTCFE534LQKR","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JTCFE534LQKRZAA7","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JTCFE534","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:64dcc124df7dc841557416e964448d8c2aaddf828e245cf6c43c860015e0e7fb","target":"graph","created_at":"2026-05-18T04:27:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In a recent paper, K. Soundararajan showed, roughly speaking, that the integers smaller than x whose prime factors are less than y are asymptotically equidistributed in arithmetic progressions to modulus q, provided that y^{4\\sqrt{e}-\\delta} \\geq q and that y is neither too large nor too small compared with x. We show that these latter restrictions on y are unnecessary, thereby proving a conjecture of Soundararajan. Our argument uses a simple majorant principle for trigonometric sums to handle a saddle point that is close to 1.","authors_text":"Adam J. Harper","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-03-10T19:03:52Z","title":"On a paper of K. Soundararajan on smooth numbers in arithmetic progressions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.2106","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2a7215c86224b2352f4caaeef29e5900b09f64bea2f5cc257e63d3190aab8f4b","target":"record","created_at":"2026-05-18T04:27:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"fd9346d1cc613d357523d997d89c7822aee1e7303d904b38bac4bd87cfd49019","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-03-10T19:03:52Z","title_canon_sha256":"8aa6de77ab6a7a8a07f8c684c529391a5531c47563ba1edac562def8836f97a9"},"schema_version":"1.0","source":{"id":"1103.2106","kind":"arxiv","version":1}},"canonical_sha256":"4cc452777c5c151c801f7490bf8165242fd55c182e6d760da475c1f214611bd4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4cc452777c5c151c801f7490bf8165242fd55c182e6d760da475c1f214611bd4","first_computed_at":"2026-05-18T04:27:02.963557Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:02.963557Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"C+0quGD/A+6ET0hV6A2zDtXbK6rCK83+JI83DVcdFwLnWJc/sqPn0+s+k/R90G7Hvcn428+NTmbC5xqZaf7hCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:02.964232Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.2106","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2a7215c86224b2352f4caaeef29e5900b09f64bea2f5cc257e63d3190aab8f4b","sha256:64dcc124df7dc841557416e964448d8c2aaddf828e245cf6c43c860015e0e7fb"],"state_sha256":"2d7ebda70aea00903075debed1c095e9997cb15d385947f6cf16f0338eda4a9d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oIBzqDm5fNX/trNb9aITfLCsv9p+lReIk8Iw1xApT/QrE5VqglQP+Xd7+BqK4JqEXYF1gRjlAf2ulqnC63BwBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T20:46:31.128281Z","bundle_sha256":"d20dfacb81a383e8e2ca6c7a0931250e3b8b5eea953cd287826033e2d33538c0"}}