{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:RZTWU42PH2WGSIP5NJ7HZ3FKSP","short_pith_number":"pith:RZTWU42P","canonical_record":{"source":{"id":"2503.09519","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-03-12T16:34:55Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"82391589c6b25e2e45e25afc4c70fe49465718d18d2de05e447614fd827a220f","abstract_canon_sha256":"c49aca0d261d7ed6adc85b22b96238d403c3c933107abf689ac6944848385cbc"},"schema_version":"1.0"},"canonical_sha256":"8e676a734f3eac6921fd6a7e7cecaa93fb889efc33250d5f2031d209877781ff","source":{"kind":"arxiv","id":"2503.09519","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2503.09519","created_at":"2026-05-22T01:03:41Z"},{"alias_kind":"arxiv_version","alias_value":"2503.09519v2","created_at":"2026-05-22T01:03:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2503.09519","created_at":"2026-05-22T01:03:41Z"},{"alias_kind":"pith_short_12","alias_value":"RZTWU42PH2WG","created_at":"2026-05-22T01:03:41Z"},{"alias_kind":"pith_short_16","alias_value":"RZTWU42PH2WGSIP5","created_at":"2026-05-22T01:03:41Z"},{"alias_kind":"pith_short_8","alias_value":"RZTWU42P","created_at":"2026-05-22T01:03:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:RZTWU42PH2WGSIP5NJ7HZ3FKSP","target":"record","payload":{"canonical_record":{"source":{"id":"2503.09519","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-03-12T16:34:55Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"82391589c6b25e2e45e25afc4c70fe49465718d18d2de05e447614fd827a220f","abstract_canon_sha256":"c49aca0d261d7ed6adc85b22b96238d403c3c933107abf689ac6944848385cbc"},"schema_version":"1.0"},"canonical_sha256":"8e676a734f3eac6921fd6a7e7cecaa93fb889efc33250d5f2031d209877781ff","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:03:41.820425Z","signature_b64":"sLq/YTIu/BRcIaOLeTye3R+fbcHjNyiD1bb9oialHU40l2NNRKi0aqK/5VAYyMkP5fxRX2JwcV/rF+PcVXP3BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e676a734f3eac6921fd6a7e7cecaa93fb889efc33250d5f2031d209877781ff","last_reissued_at":"2026-05-22T01:03:41.819585Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:03:41.819585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2503.09519","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-22T01:03:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9NwALOU4n5MHKK3NPCwOzxLsycUzN9cY/mdAIONsniTqNBrSzdQyMA4IENolplGHtqYJyMxC19bE+lhHAoMZBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T08:31:45.677660Z"},"content_sha256":"89a04622d2c6ab8a8eadcca705f5c3c0e844fa7d69545fb9101b01980a8e7208","schema_version":"1.0","event_id":"sha256:89a04622d2c6ab8a8eadcca705f5c3c0e844fa7d69545fb9101b01980a8e7208"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:RZTWU42PH2WGSIP5NJ7HZ3FKSP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Simple and accurate approximations to the Riemann zeta function","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.NT","authors_text":"Alexey Kuznetsov","submitted_at":"2025-03-12T16:34:55Z","abstract_excerpt":"We develop approximations for the Riemann zeta function that enable high-precision computation within the critical strip and other vertical strips. These approximations combine the main sum of the Riemann-Siegel formula with a simple approximation of the remainder term, which involves only elementary functions and certain precomputed coefficients obtained via Gaussian quadrature. Additionally, we provide approximations for the derivative of the Riemann zeta function and present extensive numerical evidence demonstrating the accuracy of these approximations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.09519","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2503.09519/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-05-22T01:03:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BLgw9lqZbPFHGiviLE5UYxDMMGCW6OFcY/fvWO0KnwDiRf2b6dNP3yXz0pLzqKIHPQDxjiv9wfAVXzw7TG87Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T08:31:45.678403Z"},"content_sha256":"4c81743c75a576b45d4f5901793a0691c163602d3e68e0bd86e1fff4fe610b72","schema_version":"1.0","event_id":"sha256:4c81743c75a576b45d4f5901793a0691c163602d3e68e0bd86e1fff4fe610b72"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RZTWU42PH2WGSIP5NJ7HZ3FKSP/bundle.json","state_url":"https://pith.science/pith/RZTWU42PH2WGSIP5NJ7HZ3FKSP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RZTWU42PH2WGSIP5NJ7HZ3FKSP/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":"2026-05-26T08:31:45Z","links":{"resolver":"https://pith.science/pith/RZTWU42PH2WGSIP5NJ7HZ3FKSP","bundle":"https://pith.science/pith/RZTWU42PH2WGSIP5NJ7HZ3FKSP/bundle.json","state":"https://pith.science/pith/RZTWU42PH2WGSIP5NJ7HZ3FKSP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RZTWU42PH2WGSIP5NJ7HZ3FKSP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:RZTWU42PH2WGSIP5NJ7HZ3FKSP","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":"c49aca0d261d7ed6adc85b22b96238d403c3c933107abf689ac6944848385cbc","cross_cats_sorted":["cs.NA","math.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-03-12T16:34:55Z","title_canon_sha256":"82391589c6b25e2e45e25afc4c70fe49465718d18d2de05e447614fd827a220f"},"schema_version":"1.0","source":{"id":"2503.09519","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2503.09519","created_at":"2026-05-22T01:03:41Z"},{"alias_kind":"arxiv_version","alias_value":"2503.09519v2","created_at":"2026-05-22T01:03:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2503.09519","created_at":"2026-05-22T01:03:41Z"},{"alias_kind":"pith_short_12","alias_value":"RZTWU42PH2WG","created_at":"2026-05-22T01:03:41Z"},{"alias_kind":"pith_short_16","alias_value":"RZTWU42PH2WGSIP5","created_at":"2026-05-22T01:03:41Z"},{"alias_kind":"pith_short_8","alias_value":"RZTWU42P","created_at":"2026-05-22T01:03:41Z"}],"graph_snapshots":[{"event_id":"sha256:4c81743c75a576b45d4f5901793a0691c163602d3e68e0bd86e1fff4fe610b72","target":"graph","created_at":"2026-05-22T01:03:41Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2503.09519/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We develop approximations for the Riemann zeta function that enable high-precision computation within the critical strip and other vertical strips. These approximations combine the main sum of the Riemann-Siegel formula with a simple approximation of the remainder term, which involves only elementary functions and certain precomputed coefficients obtained via Gaussian quadrature. Additionally, we provide approximations for the derivative of the Riemann zeta function and present extensive numerical evidence demonstrating the accuracy of these approximations.","authors_text":"Alexey Kuznetsov","cross_cats":["cs.NA","math.NA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-03-12T16:34:55Z","title":"Simple and accurate approximations to the Riemann zeta function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.09519","kind":"arxiv","version":2},"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":"sha256:89a04622d2c6ab8a8eadcca705f5c3c0e844fa7d69545fb9101b01980a8e7208","target":"record","created_at":"2026-05-22T01:03:41Z","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":"c49aca0d261d7ed6adc85b22b96238d403c3c933107abf689ac6944848385cbc","cross_cats_sorted":["cs.NA","math.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-03-12T16:34:55Z","title_canon_sha256":"82391589c6b25e2e45e25afc4c70fe49465718d18d2de05e447614fd827a220f"},"schema_version":"1.0","source":{"id":"2503.09519","kind":"arxiv","version":2}},"canonical_sha256":"8e676a734f3eac6921fd6a7e7cecaa93fb889efc33250d5f2031d209877781ff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8e676a734f3eac6921fd6a7e7cecaa93fb889efc33250d5f2031d209877781ff","first_computed_at":"2026-05-22T01:03:41.819585Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T01:03:41.819585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sLq/YTIu/BRcIaOLeTye3R+fbcHjNyiD1bb9oialHU40l2NNRKi0aqK/5VAYyMkP5fxRX2JwcV/rF+PcVXP3BA==","signature_status":"signed_v1","signed_at":"2026-05-22T01:03:41.820425Z","signed_message":"canonical_sha256_bytes"},"source_id":"2503.09519","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89a04622d2c6ab8a8eadcca705f5c3c0e844fa7d69545fb9101b01980a8e7208","sha256:4c81743c75a576b45d4f5901793a0691c163602d3e68e0bd86e1fff4fe610b72"],"state_sha256":"a2ebbaf495c17fd6f47a225f251c2bf6e30657af0a003da9901b44a588faced6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2JRjP3a4Yikrbdu/6yHrlLJbrlym2vS1O4LMT/7FrXycvaBwdavv1IHFn5skMY4V4J5GD9QRHnldyWsib+R6CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T08:31:45.681758Z","bundle_sha256":"88a848c5076905879a6d4df8c041fddf76c99fa941bb50ed5a1e9cece52cd30c"}}