{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:XEGLVQCCCQG4BQ4X7X5HD5LVFE","short_pith_number":"pith:XEGLVQCC","canonical_record":{"source":{"id":"1206.6188","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-27T07:16:04Z","cross_cats_sorted":[],"title_canon_sha256":"4b94273b674a232bf75570b6d82eea395eed751821cde4e4c85a9a46df778445","abstract_canon_sha256":"907d9dec496c0da41d0bee50aa7d7c6d3441a776636b597854a03a94d8e35af9"},"schema_version":"1.0"},"canonical_sha256":"b90cbac042140dc0c397fdfa71f575293c81fd623890e0a79c5843e54b81b9be","source":{"kind":"arxiv","id":"1206.6188","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.6188","created_at":"2026-05-18T03:52:27Z"},{"alias_kind":"arxiv_version","alias_value":"1206.6188v1","created_at":"2026-05-18T03:52:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.6188","created_at":"2026-05-18T03:52:27Z"},{"alias_kind":"pith_short_12","alias_value":"XEGLVQCCCQG4","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XEGLVQCCCQG4BQ4X","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XEGLVQCC","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:XEGLVQCCCQG4BQ4X7X5HD5LVFE","target":"record","payload":{"canonical_record":{"source":{"id":"1206.6188","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-27T07:16:04Z","cross_cats_sorted":[],"title_canon_sha256":"4b94273b674a232bf75570b6d82eea395eed751821cde4e4c85a9a46df778445","abstract_canon_sha256":"907d9dec496c0da41d0bee50aa7d7c6d3441a776636b597854a03a94d8e35af9"},"schema_version":"1.0"},"canonical_sha256":"b90cbac042140dc0c397fdfa71f575293c81fd623890e0a79c5843e54b81b9be","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:52:27.441209Z","signature_b64":"qkfChn1pezyF/vSAB2lhyywZMCsVCEmGOIuMmkZUwasvAJ/O1DJ2AOb1VswOsx0YhntIZx36cPGkEHdapUQlAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b90cbac042140dc0c397fdfa71f575293c81fd623890e0a79c5843e54b81b9be","last_reissued_at":"2026-05-18T03:52:27.440442Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:52:27.440442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.6188","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":"2026-05-18T03:52:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YfZ/jlQK7Fmx2LMvqverD8BRMJJybPHPQCSifxVeKWBshDrVe18dTJKA7EDiG7PwKyUGgszHzrY3DXbWkjJyBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T14:44:04.040339Z"},"content_sha256":"fdc99cd23d3343186d8d6e1fb19b49050821b4b6b6faf1de5216ee934d72313b","schema_version":"1.0","event_id":"sha256:fdc99cd23d3343186d8d6e1fb19b49050821b4b6b6faf1de5216ee934d72313b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:XEGLVQCCCQG4BQ4X7X5HD5LVFE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ferenc Moricz","submitted_at":"2012-06-27T07:16:04Z","abstract_excerpt":"Let $s: [1, \\infty) \\to \\C$ be a locally integrable function in Lebesgue's sense on the infinite interval $[1, \\infty)$. We say that $s$ is summable $(L, 1)$ if there exists some $A\\in \\C$ such that $$\\lim_{t\\to \\infty} \\tau(t) = A, \\quad {\\rm where} \\quad \\tau(t):= {1\\over \\log t} \\int^t_1 {s(u) \\over u} du.\\leqno(*)$$ It is clear that if the ordinary limit $s(t) \\to A$ exists, then the limit $\\tau(t) \\to A$ also exists as $t\\to \\infty$. We present sufficient conditions, which are also necessary in order that the converse implication hold true. As corollaries, we obtain so-called Tauberian th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.6188","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":"2026-05-18T03:52:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ub9e6hyBsR28dCRWUbXMsPOKvFwvh3rHnlcwAL/gjR5NBqwRqlP5tF77OEajv0Rr5g6mZBuNax1cL3jc3UhtDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T14:44:04.040681Z"},"content_sha256":"744611f73dc72d4ad22b1dd63e22bb9cb7ab2dcf350263f7d70f6e13b365e34e","schema_version":"1.0","event_id":"sha256:744611f73dc72d4ad22b1dd63e22bb9cb7ab2dcf350263f7d70f6e13b365e34e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE/bundle.json","state_url":"https://pith.science/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE/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":"2026-06-07T14:44:04Z","links":{"resolver":"https://pith.science/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE","bundle":"https://pith.science/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE/bundle.json","state":"https://pith.science/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:XEGLVQCCCQG4BQ4X7X5HD5LVFE","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":"907d9dec496c0da41d0bee50aa7d7c6d3441a776636b597854a03a94d8e35af9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-27T07:16:04Z","title_canon_sha256":"4b94273b674a232bf75570b6d82eea395eed751821cde4e4c85a9a46df778445"},"schema_version":"1.0","source":{"id":"1206.6188","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.6188","created_at":"2026-05-18T03:52:27Z"},{"alias_kind":"arxiv_version","alias_value":"1206.6188v1","created_at":"2026-05-18T03:52:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.6188","created_at":"2026-05-18T03:52:27Z"},{"alias_kind":"pith_short_12","alias_value":"XEGLVQCCCQG4","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XEGLVQCCCQG4BQ4X","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XEGLVQCC","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:744611f73dc72d4ad22b1dd63e22bb9cb7ab2dcf350263f7d70f6e13b365e34e","target":"graph","created_at":"2026-05-18T03:52:27Z","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":"Let $s: [1, \\infty) \\to \\C$ be a locally integrable function in Lebesgue's sense on the infinite interval $[1, \\infty)$. We say that $s$ is summable $(L, 1)$ if there exists some $A\\in \\C$ such that $$\\lim_{t\\to \\infty} \\tau(t) = A, \\quad {\\rm where} \\quad \\tau(t):= {1\\over \\log t} \\int^t_1 {s(u) \\over u} du.\\leqno(*)$$ It is clear that if the ordinary limit $s(t) \\to A$ exists, then the limit $\\tau(t) \\to A$ also exists as $t\\to \\infty$. We present sufficient conditions, which are also necessary in order that the converse implication hold true. As corollaries, we obtain so-called Tauberian th","authors_text":"Ferenc Moricz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-27T07:16:04Z","title":"Necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.6188","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:fdc99cd23d3343186d8d6e1fb19b49050821b4b6b6faf1de5216ee934d72313b","target":"record","created_at":"2026-05-18T03:52:27Z","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":"907d9dec496c0da41d0bee50aa7d7c6d3441a776636b597854a03a94d8e35af9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-27T07:16:04Z","title_canon_sha256":"4b94273b674a232bf75570b6d82eea395eed751821cde4e4c85a9a46df778445"},"schema_version":"1.0","source":{"id":"1206.6188","kind":"arxiv","version":1}},"canonical_sha256":"b90cbac042140dc0c397fdfa71f575293c81fd623890e0a79c5843e54b81b9be","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b90cbac042140dc0c397fdfa71f575293c81fd623890e0a79c5843e54b81b9be","first_computed_at":"2026-05-18T03:52:27.440442Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:52:27.440442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qkfChn1pezyF/vSAB2lhyywZMCsVCEmGOIuMmkZUwasvAJ/O1DJ2AOb1VswOsx0YhntIZx36cPGkEHdapUQlAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:52:27.441209Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.6188","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fdc99cd23d3343186d8d6e1fb19b49050821b4b6b6faf1de5216ee934d72313b","sha256:744611f73dc72d4ad22b1dd63e22bb9cb7ab2dcf350263f7d70f6e13b365e34e"],"state_sha256":"0b852d5cd32493eef09a7cf411dd254c70c46cfd56ec6ccc69139549b9404ac6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iYOVR9RVXqoW4LEOOJ9tS1Oq1QZs4ebco/GAP8kx2OpfXwztKCnsoKUVTDsmjBw9pcmA68gIqTP7SXC94XBgDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T14:44:04.042537Z","bundle_sha256":"46c27caa5eff29e8607da21aa783e374a35982a5ed7af9d38d979239d9e73e54"}}