{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:2YTXFEGSATL7MSHLI2V4M55C4H","short_pith_number":"pith:2YTXFEGS","canonical_record":{"source":{"id":"1905.00009","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-04-30T16:31:50Z","cross_cats_sorted":[],"title_canon_sha256":"b055b29e4ff5463bfaef4f9a81922b2de1c7127ff0416bbe731ddadc90f5331c","abstract_canon_sha256":"b9ca77fb628798ba52e4200c630d324b71d8a4179cb8bf31cb55290889fddd57"},"schema_version":"1.0"},"canonical_sha256":"d6277290d204d7f648eb46abc677a2e1e66b4073b7da2f5b2411af86701c3ee1","source":{"kind":"arxiv","id":"1905.00009","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.00009","created_at":"2026-05-17T23:47:15Z"},{"alias_kind":"arxiv_version","alias_value":"1905.00009v1","created_at":"2026-05-17T23:47:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.00009","created_at":"2026-05-17T23:47:15Z"},{"alias_kind":"pith_short_12","alias_value":"2YTXFEGSATL7","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"2YTXFEGSATL7MSHL","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"2YTXFEGS","created_at":"2026-05-18T12:33:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:2YTXFEGSATL7MSHLI2V4M55C4H","target":"record","payload":{"canonical_record":{"source":{"id":"1905.00009","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-04-30T16:31:50Z","cross_cats_sorted":[],"title_canon_sha256":"b055b29e4ff5463bfaef4f9a81922b2de1c7127ff0416bbe731ddadc90f5331c","abstract_canon_sha256":"b9ca77fb628798ba52e4200c630d324b71d8a4179cb8bf31cb55290889fddd57"},"schema_version":"1.0"},"canonical_sha256":"d6277290d204d7f648eb46abc677a2e1e66b4073b7da2f5b2411af86701c3ee1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:15.336934Z","signature_b64":"Kuc8TqyKDn6tROoJlhPAzo6IQbltE27tFlJVAeDOim4htjtaeL3Mi52sTkOPWDJ1KwDtMevxrSNbp3ff9FJ2Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d6277290d204d7f648eb46abc677a2e1e66b4073b7da2f5b2411af86701c3ee1","last_reissued_at":"2026-05-17T23:47:15.336521Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:15.336521Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1905.00009","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":"2026-05-17T23:47:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"t7tbmP1veJC5BGOrxtS39GcME/D8SxiU9SIBm0yAs7d9O9g+2vA4JpDgFpBk9EJRtGX1HXTmW3y2/qyzI40wBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:13:55.860894Z"},"content_sha256":"b7e088072688ab7eca91fc9d058cb255211dde8b31812840c32137c96c5928c8","schema_version":"1.0","event_id":"sha256:b7e088072688ab7eca91fc9d058cb255211dde8b31812840c32137c96c5928c8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:2YTXFEGSATL7MSHLI2V4M55C4H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotics of a sum of modified Bessel functions with non-linear argument","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"R B Paris","submitted_at":"2019-04-30T16:31:50Z","abstract_excerpt":"We examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by \\[S_{\\nu,p}(a)=\\sum_{n\\geq 1} (an^p/2)^{-\\nu} K_\\nu(an^p)\\qquad (a>0,\\ 0\\leq\\nu<1)\\] as the parameter $a\\to 0+$, where $p$ denotes an integer satisfying $p\\geq 2$. This extends previous work for the cases $p=1$ (linear) and $p=2$ (quadratic). The expansion as $a\\to0+$ consists of an infinite number of asymptotic sums involving the Riemann zeta function, which when optimally truncated lead to remainder terms that are exponentially small in the parameter $a$. The number of these e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.00009","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":"2026-05-17T23:47:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MDM7O/GLDxFEu7i3HOVsdQAlVMnsnMnBKPiYOKBTcWmJM8N32+Ng+VH1E7kQD0UYoKU7rJK4VzFe2cXSQzRtBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:13:55.861663Z"},"content_sha256":"7541d6226505c01ea5b34d32811de35328f0d2bb2b2e30bbdf508a0aaaf4e882","schema_version":"1.0","event_id":"sha256:7541d6226505c01ea5b34d32811de35328f0d2bb2b2e30bbdf508a0aaaf4e882"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2YTXFEGSATL7MSHLI2V4M55C4H/bundle.json","state_url":"https://pith.science/pith/2YTXFEGSATL7MSHLI2V4M55C4H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2YTXFEGSATL7MSHLI2V4M55C4H/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":"2026-06-11T23:13:55Z","links":{"resolver":"https://pith.science/pith/2YTXFEGSATL7MSHLI2V4M55C4H","bundle":"https://pith.science/pith/2YTXFEGSATL7MSHLI2V4M55C4H/bundle.json","state":"https://pith.science/pith/2YTXFEGSATL7MSHLI2V4M55C4H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2YTXFEGSATL7MSHLI2V4M55C4H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:2YTXFEGSATL7MSHLI2V4M55C4H","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":"b9ca77fb628798ba52e4200c630d324b71d8a4179cb8bf31cb55290889fddd57","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-04-30T16:31:50Z","title_canon_sha256":"b055b29e4ff5463bfaef4f9a81922b2de1c7127ff0416bbe731ddadc90f5331c"},"schema_version":"1.0","source":{"id":"1905.00009","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.00009","created_at":"2026-05-17T23:47:15Z"},{"alias_kind":"arxiv_version","alias_value":"1905.00009v1","created_at":"2026-05-17T23:47:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.00009","created_at":"2026-05-17T23:47:15Z"},{"alias_kind":"pith_short_12","alias_value":"2YTXFEGSATL7","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"2YTXFEGSATL7MSHL","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"2YTXFEGS","created_at":"2026-05-18T12:33:07Z"}],"graph_snapshots":[{"event_id":"sha256:7541d6226505c01ea5b34d32811de35328f0d2bb2b2e30bbdf508a0aaaf4e882","target":"graph","created_at":"2026-05-17T23:47:15Z","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 examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by \\[S_{\\nu,p}(a)=\\sum_{n\\geq 1} (an^p/2)^{-\\nu} K_\\nu(an^p)\\qquad (a>0,\\ 0\\leq\\nu<1)\\] as the parameter $a\\to 0+$, where $p$ denotes an integer satisfying $p\\geq 2$. This extends previous work for the cases $p=1$ (linear) and $p=2$ (quadratic). The expansion as $a\\to0+$ consists of an infinite number of asymptotic sums involving the Riemann zeta function, which when optimally truncated lead to remainder terms that are exponentially small in the parameter $a$. The number of these e","authors_text":"R B Paris","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-04-30T16:31:50Z","title":"Asymptotics of a sum of modified Bessel functions with non-linear argument"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.00009","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:b7e088072688ab7eca91fc9d058cb255211dde8b31812840c32137c96c5928c8","target":"record","created_at":"2026-05-17T23:47:15Z","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":"b9ca77fb628798ba52e4200c630d324b71d8a4179cb8bf31cb55290889fddd57","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-04-30T16:31:50Z","title_canon_sha256":"b055b29e4ff5463bfaef4f9a81922b2de1c7127ff0416bbe731ddadc90f5331c"},"schema_version":"1.0","source":{"id":"1905.00009","kind":"arxiv","version":1}},"canonical_sha256":"d6277290d204d7f648eb46abc677a2e1e66b4073b7da2f5b2411af86701c3ee1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d6277290d204d7f648eb46abc677a2e1e66b4073b7da2f5b2411af86701c3ee1","first_computed_at":"2026-05-17T23:47:15.336521Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:47:15.336521Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Kuc8TqyKDn6tROoJlhPAzo6IQbltE27tFlJVAeDOim4htjtaeL3Mi52sTkOPWDJ1KwDtMevxrSNbp3ff9FJ2Cg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:47:15.336934Z","signed_message":"canonical_sha256_bytes"},"source_id":"1905.00009","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b7e088072688ab7eca91fc9d058cb255211dde8b31812840c32137c96c5928c8","sha256:7541d6226505c01ea5b34d32811de35328f0d2bb2b2e30bbdf508a0aaaf4e882"],"state_sha256":"b59380ff9a13e9219b093e2ca659203bb31ac9a290de5939aa34b1a0464ec580"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zKMLPRlwXevrZO9EAm0F5PFQWvRYCVxlSxZkRaEEnM4XwkHb0byO1ok7E0Z7I/Dw+YrKkS+13GpEhSf4DVvNCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T23:13:55.865353Z","bundle_sha256":"342520357a63fb9813d15d1c21e1d81339f1e1ad0546c5c1583e37d3af1a5b42"}}