{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2003:34I2DDVGJBPUZMWZVJDSSAWNF6","short_pith_number":"pith:34I2DDVG","canonical_record":{"source":{"id":"math/0308009","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2003-08-01T18:34:07Z","cross_cats_sorted":[],"title_canon_sha256":"2c417bb4cbcedb496c388b320336cbfc35e40f5fb38246d487a82d3536ff65cc","abstract_canon_sha256":"dbd3a5c435cf6577c6f40d7639960a8e72a0fc9c5c22bcb8d869929265e77e37"},"schema_version":"1.0"},"canonical_sha256":"df11a18ea6485f4cb2d9aa472902cd2fbfd951d8258c4c3b211aecd6153a7130","source":{"kind":"arxiv","id":"math/0308009","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0308009","created_at":"2026-05-18T01:38:28Z"},{"alias_kind":"arxiv_version","alias_value":"math/0308009v1","created_at":"2026-05-18T01:38:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0308009","created_at":"2026-05-18T01:38:28Z"},{"alias_kind":"pith_short_12","alias_value":"34I2DDVGJBPU","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_16","alias_value":"34I2DDVGJBPUZMWZ","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_8","alias_value":"34I2DDVG","created_at":"2026-05-18T12:25:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2003:34I2DDVGJBPUZMWZVJDSSAWNF6","target":"record","payload":{"canonical_record":{"source":{"id":"math/0308009","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2003-08-01T18:34:07Z","cross_cats_sorted":[],"title_canon_sha256":"2c417bb4cbcedb496c388b320336cbfc35e40f5fb38246d487a82d3536ff65cc","abstract_canon_sha256":"dbd3a5c435cf6577c6f40d7639960a8e72a0fc9c5c22bcb8d869929265e77e37"},"schema_version":"1.0"},"canonical_sha256":"df11a18ea6485f4cb2d9aa472902cd2fbfd951d8258c4c3b211aecd6153a7130","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:28.907458Z","signature_b64":"/uIR5Zz0G+IXw9frqPrmB7Zga2QvmUZIfTwpElBNKRRaJNiOiX/WY15LdGHd3VOWgJNCtk3FlDmKGzy1wFuUDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"df11a18ea6485f4cb2d9aa472902cd2fbfd951d8258c4c3b211aecd6153a7130","last_reissued_at":"2026-05-18T01:38:28.906728Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:28.906728Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0308009","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":"2026-05-18T01:38:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6jZVSjY4nxjykwNAXrQjYddAK8ShvF3EPUr9rhCmEOw18iF0/y4f8BQYyO6uefWOs+x6VJbLcPRDpCmvPYM5Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T16:31:07.059163Z"},"content_sha256":"fb740ec88e6d7802bb9f647dca05bf14467942d70aadf90229741304079a3cb8","schema_version":"1.0","event_id":"sha256:fb740ec88e6d7802bb9f647dca05bf14467942d70aadf90229741304079a3cb8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2003:34I2DDVGJBPUZMWZVJDSSAWNF6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The number of unsieved integers up to x","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrew Granville, Kannan Soundararajan","submitted_at":"2003-08-01T18:34:07Z","abstract_excerpt":"Typically, one expects that there are around x\\prod_{p\\not\\in P, p <= x} (1-1/p) integers up to x, all of whose prime factors come from the set P. Of course for some choices of P one may get rather more integers, and for some choices of P one may get rather less. Hall [4] showed that one never gets more than e^\\gamma+o(1) times the expected amount (where \\gamma is the Euler-Mascheroni constant), which was improved slightly by Hildebrand [5]. Hildebrand [6] also showed that for a given value of \\prod_{p\\not\\in P, p <= x} (1-1/p), the smallest count that you get (asymptotically) is when P consis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0308009","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":"2026-05-18T01:38:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N2H+evwoVGq8Vt8WvZ6+AKiP2OexiMTQBvPZRV5xR57ZaZ3CV0WF4+xQEzSaoo339dEnPbK/XDU+6XdreYWQAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T16:31:07.059526Z"},"content_sha256":"75b6802113601aa554b4aa7bf052b58a838bc70b010e4ec5d9bd91595a072e59","schema_version":"1.0","event_id":"sha256:75b6802113601aa554b4aa7bf052b58a838bc70b010e4ec5d9bd91595a072e59"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/34I2DDVGJBPUZMWZVJDSSAWNF6/bundle.json","state_url":"https://pith.science/pith/34I2DDVGJBPUZMWZVJDSSAWNF6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/34I2DDVGJBPUZMWZVJDSSAWNF6/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":"2026-05-30T16:31:07Z","links":{"resolver":"https://pith.science/pith/34I2DDVGJBPUZMWZVJDSSAWNF6","bundle":"https://pith.science/pith/34I2DDVGJBPUZMWZVJDSSAWNF6/bundle.json","state":"https://pith.science/pith/34I2DDVGJBPUZMWZVJDSSAWNF6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/34I2DDVGJBPUZMWZVJDSSAWNF6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2003:34I2DDVGJBPUZMWZVJDSSAWNF6","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":"dbd3a5c435cf6577c6f40d7639960a8e72a0fc9c5c22bcb8d869929265e77e37","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2003-08-01T18:34:07Z","title_canon_sha256":"2c417bb4cbcedb496c388b320336cbfc35e40f5fb38246d487a82d3536ff65cc"},"schema_version":"1.0","source":{"id":"math/0308009","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0308009","created_at":"2026-05-18T01:38:28Z"},{"alias_kind":"arxiv_version","alias_value":"math/0308009v1","created_at":"2026-05-18T01:38:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0308009","created_at":"2026-05-18T01:38:28Z"},{"alias_kind":"pith_short_12","alias_value":"34I2DDVGJBPU","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_16","alias_value":"34I2DDVGJBPUZMWZ","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_8","alias_value":"34I2DDVG","created_at":"2026-05-18T12:25:51Z"}],"graph_snapshots":[{"event_id":"sha256:75b6802113601aa554b4aa7bf052b58a838bc70b010e4ec5d9bd91595a072e59","target":"graph","created_at":"2026-05-18T01:38:28Z","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":"Typically, one expects that there are around x\\prod_{p\\not\\in P, p <= x} (1-1/p) integers up to x, all of whose prime factors come from the set P. Of course for some choices of P one may get rather more integers, and for some choices of P one may get rather less. Hall [4] showed that one never gets more than e^\\gamma+o(1) times the expected amount (where \\gamma is the Euler-Mascheroni constant), which was improved slightly by Hildebrand [5]. Hildebrand [6] also showed that for a given value of \\prod_{p\\not\\in P, p <= x} (1-1/p), the smallest count that you get (asymptotically) is when P consis","authors_text":"Andrew Granville, Kannan Soundararajan","cross_cats":[],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"2003-08-01T18:34:07Z","title":"The number of unsieved integers up to x"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0308009","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:fb740ec88e6d7802bb9f647dca05bf14467942d70aadf90229741304079a3cb8","target":"record","created_at":"2026-05-18T01:38:28Z","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":"dbd3a5c435cf6577c6f40d7639960a8e72a0fc9c5c22bcb8d869929265e77e37","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2003-08-01T18:34:07Z","title_canon_sha256":"2c417bb4cbcedb496c388b320336cbfc35e40f5fb38246d487a82d3536ff65cc"},"schema_version":"1.0","source":{"id":"math/0308009","kind":"arxiv","version":1}},"canonical_sha256":"df11a18ea6485f4cb2d9aa472902cd2fbfd951d8258c4c3b211aecd6153a7130","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"df11a18ea6485f4cb2d9aa472902cd2fbfd951d8258c4c3b211aecd6153a7130","first_computed_at":"2026-05-18T01:38:28.906728Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:38:28.906728Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/uIR5Zz0G+IXw9frqPrmB7Zga2QvmUZIfTwpElBNKRRaJNiOiX/WY15LdGHd3VOWgJNCtk3FlDmKGzy1wFuUDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:38:28.907458Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0308009","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb740ec88e6d7802bb9f647dca05bf14467942d70aadf90229741304079a3cb8","sha256:75b6802113601aa554b4aa7bf052b58a838bc70b010e4ec5d9bd91595a072e59"],"state_sha256":"f3556ac26524b417b610baa4aae4b8709337889190eb162f7eefb2d28a81db41"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LaW+zkqR5PnY0xc6ntGhnOepX9KlYpEmTHUUmsP6nLXX85ly4Bo0/gDmb1L2zC3uDionXpipyIbJ9fxmqQ4ICQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T16:31:07.061574Z","bundle_sha256":"33f130adb8120e744c00b056e1c80dbc3d859921d4c3f7689ae611a7932197de"}}