{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:5OLLC5TNNLHPMDNEUT6RLLVYEJ","short_pith_number":"pith:5OLLC5TN","canonical_record":{"source":{"id":"1107.2007","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-11T12:09:57Z","cross_cats_sorted":[],"title_canon_sha256":"cb95e43dae26df87fb30a1f55883015381c96bd299422fabb34a68cd60f6e722","abstract_canon_sha256":"5a858022f7426e0028ab366f71a4b8cee35aea16b8a7c8c937f6237932fccaba"},"schema_version":"1.0"},"canonical_sha256":"eb96b1766d6acef60da4a4fd15aeb8224419373ef9aa60ca0b99af37d1416276","source":{"kind":"arxiv","id":"1107.2007","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2007","created_at":"2026-05-18T04:18:21Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2007v2","created_at":"2026-05-18T04:18:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2007","created_at":"2026-05-18T04:18:21Z"},{"alias_kind":"pith_short_12","alias_value":"5OLLC5TNNLHP","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"5OLLC5TNNLHPMDNE","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"5OLLC5TN","created_at":"2026-05-18T12:26:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:5OLLC5TNNLHPMDNEUT6RLLVYEJ","target":"record","payload":{"canonical_record":{"source":{"id":"1107.2007","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-11T12:09:57Z","cross_cats_sorted":[],"title_canon_sha256":"cb95e43dae26df87fb30a1f55883015381c96bd299422fabb34a68cd60f6e722","abstract_canon_sha256":"5a858022f7426e0028ab366f71a4b8cee35aea16b8a7c8c937f6237932fccaba"},"schema_version":"1.0"},"canonical_sha256":"eb96b1766d6acef60da4a4fd15aeb8224419373ef9aa60ca0b99af37d1416276","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:21.968993Z","signature_b64":"iFVnLKQoJoRXtYKrVMlUvpYwLEsdNIxrntfArSR6vdQQ7F7qVkD4p8MkNPUbVneD0B0KztAqveHOOi2TGL0FCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eb96b1766d6acef60da4a4fd15aeb8224419373ef9aa60ca0b99af37d1416276","last_reissued_at":"2026-05-18T04:18:21.968552Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:21.968552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.2007","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":"2026-05-18T04:18:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qqhxqZhugKHCWsZJP040UIGfMNJt1d8fGetuTK44W3P6DnBCWbbCywfoqAfO4IOMabL9PkyVTssBmer6XVa4Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T07:21:25.591815Z"},"content_sha256":"399b7e627963af99b90603d5a9a2b3cf8b86a3cf97c839e015fefb438201cb21","schema_version":"1.0","event_id":"sha256:399b7e627963af99b90603d5a9a2b3cf8b86a3cf97c839e015fefb438201cb21"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:5OLLC5TNNLHPMDNEUT6RLLVYEJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some asymptotics for the Bessel functions with an explicit error term","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ilia Krasikov","submitted_at":"2011-07-11T12:09:57Z","abstract_excerpt":"We show how one can obtain an asymptotic expression for some special functions satisfying a second order differential equation with a very explicit error term starting from appropriate upper bounds.\n  We will work out the details for the Bessel function $J_\\nu (x)$ and the Airy function $Ai(x)$ and find a sharp approximation for their zeros.\n  We also answer the question raised by Olenko by showing that\n  $$c_1 | \\nu^2-1/4\\,| < \\sup_{x \\ge 0} x^{3/2}|J_\\nu(x)-\\sqrt{\\frac{2}{\\pi x}} \\, \\cos (x-\\frac{\\pi \\nu}{2}-\\frac{\\pi}{4}\\,)| <c_2 |\\nu^2-1/4\\,|, $$ $ \\nu \\ge -1/2 \\, ,$ for some explicit nume"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2007","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:18:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gF2OWJj4e9AB69FuGfjq1CPJSFBCfhttz0PdXSbPPgv39E8oWcCvkxA6dsWesYkNg8DlExbf4R5e+H6FXTO2Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T07:21:25.592675Z"},"content_sha256":"797ca7c21c007d9624ae3c579ab28ecad44c5fdd79d2a431ea871705c8811e33","schema_version":"1.0","event_id":"sha256:797ca7c21c007d9624ae3c579ab28ecad44c5fdd79d2a431ea871705c8811e33"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5OLLC5TNNLHPMDNEUT6RLLVYEJ/bundle.json","state_url":"https://pith.science/pith/5OLLC5TNNLHPMDNEUT6RLLVYEJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5OLLC5TNNLHPMDNEUT6RLLVYEJ/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":"2026-08-09T07:21:25Z","links":{"resolver":"https://pith.science/pith/5OLLC5TNNLHPMDNEUT6RLLVYEJ","bundle":"https://pith.science/pith/5OLLC5TNNLHPMDNEUT6RLLVYEJ/bundle.json","state":"https://pith.science/pith/5OLLC5TNNLHPMDNEUT6RLLVYEJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5OLLC5TNNLHPMDNEUT6RLLVYEJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:5OLLC5TNNLHPMDNEUT6RLLVYEJ","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":"5a858022f7426e0028ab366f71a4b8cee35aea16b8a7c8c937f6237932fccaba","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-11T12:09:57Z","title_canon_sha256":"cb95e43dae26df87fb30a1f55883015381c96bd299422fabb34a68cd60f6e722"},"schema_version":"1.0","source":{"id":"1107.2007","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2007","created_at":"2026-05-18T04:18:21Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2007v2","created_at":"2026-05-18T04:18:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2007","created_at":"2026-05-18T04:18:21Z"},{"alias_kind":"pith_short_12","alias_value":"5OLLC5TNNLHP","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"5OLLC5TNNLHPMDNE","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"5OLLC5TN","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:797ca7c21c007d9624ae3c579ab28ecad44c5fdd79d2a431ea871705c8811e33","target":"graph","created_at":"2026-05-18T04:18:21Z","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":"We show how one can obtain an asymptotic expression for some special functions satisfying a second order differential equation with a very explicit error term starting from appropriate upper bounds.\n  We will work out the details for the Bessel function $J_\\nu (x)$ and the Airy function $Ai(x)$ and find a sharp approximation for their zeros.\n  We also answer the question raised by Olenko by showing that\n  $$c_1 | \\nu^2-1/4\\,| < \\sup_{x \\ge 0} x^{3/2}|J_\\nu(x)-\\sqrt{\\frac{2}{\\pi x}} \\, \\cos (x-\\frac{\\pi \\nu}{2}-\\frac{\\pi}{4}\\,)| <c_2 |\\nu^2-1/4\\,|, $$ $ \\nu \\ge -1/2 \\, ,$ for some explicit nume","authors_text":"Ilia Krasikov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-11T12:09:57Z","title":"Some asymptotics for the Bessel functions with an explicit error term"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2007","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":"sha256:399b7e627963af99b90603d5a9a2b3cf8b86a3cf97c839e015fefb438201cb21","target":"record","created_at":"2026-05-18T04:18:21Z","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":"5a858022f7426e0028ab366f71a4b8cee35aea16b8a7c8c937f6237932fccaba","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-11T12:09:57Z","title_canon_sha256":"cb95e43dae26df87fb30a1f55883015381c96bd299422fabb34a68cd60f6e722"},"schema_version":"1.0","source":{"id":"1107.2007","kind":"arxiv","version":2}},"canonical_sha256":"eb96b1766d6acef60da4a4fd15aeb8224419373ef9aa60ca0b99af37d1416276","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eb96b1766d6acef60da4a4fd15aeb8224419373ef9aa60ca0b99af37d1416276","first_computed_at":"2026-05-18T04:18:21.968552Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:18:21.968552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iFVnLKQoJoRXtYKrVMlUvpYwLEsdNIxrntfArSR6vdQQ7F7qVkD4p8MkNPUbVneD0B0KztAqveHOOi2TGL0FCw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:18:21.968993Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.2007","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:399b7e627963af99b90603d5a9a2b3cf8b86a3cf97c839e015fefb438201cb21","sha256:797ca7c21c007d9624ae3c579ab28ecad44c5fdd79d2a431ea871705c8811e33"],"state_sha256":"66149e5b403aca7226ce8046b740ce39378cd30c2dadb66f587eb7a93bec5f72"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y2rGW0cS4dTKYcprA1I19Y6Khoo5WVJaoxwjyow8XOeZrFL8V7kFz6g6o40ez+xReXc0UzJaDqB1JBsschwdBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T07:21:25.603577Z","bundle_sha256":"19440d0b32938ebe92c226f6dc27a70e151e8d7d6b33849a8a13e8980277f59e"}}