{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:XEGLVQCCCQG4BQ4X7X5HD5LVFE","short_pith_number":"pith:XEGLVQCC","schema_version":"1.0","canonical_sha256":"b90cbac042140dc0c397fdfa71f575293c81fd623890e0a79c5843e54b81b9be","source":{"kind":"arxiv","id":"1206.6188","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1206.6188","created_at":"2026-05-18T03:52:27.440587+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.6188v1","created_at":"2026-05-18T03:52:27.440587+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.6188","created_at":"2026-05-18T03:52:27.440587+00:00"},{"alias_kind":"pith_short_12","alias_value":"XEGLVQCCCQG4","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_16","alias_value":"XEGLVQCCCQG4BQ4X","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_8","alias_value":"XEGLVQCC","created_at":"2026-05-18T12:27:27.928770+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE","json":"https://pith.science/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE.json","graph_json":"https://pith.science/api/pith-number/XEGLVQCCCQG4BQ4X7X5HD5LVFE/graph.json","events_json":"https://pith.science/api/pith-number/XEGLVQCCCQG4BQ4X7X5HD5LVFE/events.json","paper":"https://pith.science/paper/XEGLVQCC"},"agent_actions":{"view_html":"https://pith.science/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE","download_json":"https://pith.science/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE.json","view_paper":"https://pith.science/paper/XEGLVQCC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.6188&json=true","fetch_graph":"https://pith.science/api/pith-number/XEGLVQCCCQG4BQ4X7X5HD5LVFE/graph.json","fetch_events":"https://pith.science/api/pith-number/XEGLVQCCCQG4BQ4X7X5HD5LVFE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE/action/storage_attestation","attest_author":"https://pith.science/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE/action/author_attestation","sign_citation":"https://pith.science/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE/action/citation_signature","submit_replication":"https://pith.science/pith/XEGLVQCCCQG4BQ4X7X5HD5LVFE/action/replication_record"}},"created_at":"2026-05-18T03:52:27.440587+00:00","updated_at":"2026-05-18T03:52:27.440587+00:00"}