{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:QYYA4LTUJE5KUVGTSKOCKKLINT","short_pith_number":"pith:QYYA4LTU","schema_version":"1.0","canonical_sha256":"86300e2e74493aaa54d3929c2529686cd8ce70ab3709a1eab14dd42368515a2d","source":{"kind":"arxiv","id":"1207.5366","version":1},"attestation_state":"computed","paper":{"title":"Restricted Sum Formula of Alternating Euler Sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jianqiang Zhao","submitted_at":"2012-07-23T12:02:43Z","abstract_excerpt":"In this paper we study restricted sum formulas involving alternating Euler sums which are defined by\n  \\zeta(s_1,...,s_{d};\\epsilon_1,...,\\epsilon_d)=\\sum_{n_1>...>n_d\\ge 1}\\frac{\\epsilon_1^{n_1}... \\epsilon_{d}^{n_d}}{n_1^{s_1}... n_d^{s_d}},\nfor all positive integers s_1,...,s_{d} and \\epsilon_1=\\pm 1,..., \\epsilon_{d}=\\pm 1 with (s_1,\\epsilon_1) unequal (1,1). We call w=s_1+...+s_{d} the weight and d the depth. When \\epsilon_j=-1 we say the jth component is alternating. We first consider Euler sums of the following special type:\n  \\xi(2s_1,...,2s_{d})=\\zeta(2s_1,...,2s_{d};(-1)^{s_1},...,(-"},"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":"1207.5366","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-07-23T12:02:43Z","cross_cats_sorted":[],"title_canon_sha256":"056bb4e38ccce0cd4d4420cce966a299ee5e8307ea48953679285eb6fe32fcbb","abstract_canon_sha256":"d2169aa757764a9a8100ac031c4a820d4a8b3bddeadc806c0e5c261ecd10892b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:21.232530Z","signature_b64":"tA2B/MH6L4BuBlKP5NIB36RTIPOSJ3DXZAfVqRfo/Z6K9o32IUC3PdFXlENm4aHcWKWFrHKymLf5ik4qV/b8Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"86300e2e74493aaa54d3929c2529686cd8ce70ab3709a1eab14dd42368515a2d","last_reissued_at":"2026-05-18T02:28:21.231921Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:21.231921Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Restricted Sum Formula of Alternating Euler Sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jianqiang Zhao","submitted_at":"2012-07-23T12:02:43Z","abstract_excerpt":"In this paper we study restricted sum formulas involving alternating Euler sums which are defined by\n  \\zeta(s_1,...,s_{d};\\epsilon_1,...,\\epsilon_d)=\\sum_{n_1>...>n_d\\ge 1}\\frac{\\epsilon_1^{n_1}... \\epsilon_{d}^{n_d}}{n_1^{s_1}... n_d^{s_d}},\nfor all positive integers s_1,...,s_{d} and \\epsilon_1=\\pm 1,..., \\epsilon_{d}=\\pm 1 with (s_1,\\epsilon_1) unequal (1,1). We call w=s_1+...+s_{d} the weight and d the depth. When \\epsilon_j=-1 we say the jth component is alternating. We first consider Euler sums of the following special type:\n  \\xi(2s_1,...,2s_{d})=\\zeta(2s_1,...,2s_{d};(-1)^{s_1},...,(-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5366","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":"1207.5366","created_at":"2026-05-18T02:28:21.232009+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.5366v1","created_at":"2026-05-18T02:28:21.232009+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.5366","created_at":"2026-05-18T02:28:21.232009+00:00"},{"alias_kind":"pith_short_12","alias_value":"QYYA4LTUJE5K","created_at":"2026-05-18T12:27:20.899486+00:00"},{"alias_kind":"pith_short_16","alias_value":"QYYA4LTUJE5KUVGT","created_at":"2026-05-18T12:27:20.899486+00:00"},{"alias_kind":"pith_short_8","alias_value":"QYYA4LTU","created_at":"2026-05-18T12:27:20.899486+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/QYYA4LTUJE5KUVGTSKOCKKLINT","json":"https://pith.science/pith/QYYA4LTUJE5KUVGTSKOCKKLINT.json","graph_json":"https://pith.science/api/pith-number/QYYA4LTUJE5KUVGTSKOCKKLINT/graph.json","events_json":"https://pith.science/api/pith-number/QYYA4LTUJE5KUVGTSKOCKKLINT/events.json","paper":"https://pith.science/paper/QYYA4LTU"},"agent_actions":{"view_html":"https://pith.science/pith/QYYA4LTUJE5KUVGTSKOCKKLINT","download_json":"https://pith.science/pith/QYYA4LTUJE5KUVGTSKOCKKLINT.json","view_paper":"https://pith.science/paper/QYYA4LTU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.5366&json=true","fetch_graph":"https://pith.science/api/pith-number/QYYA4LTUJE5KUVGTSKOCKKLINT/graph.json","fetch_events":"https://pith.science/api/pith-number/QYYA4LTUJE5KUVGTSKOCKKLINT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QYYA4LTUJE5KUVGTSKOCKKLINT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QYYA4LTUJE5KUVGTSKOCKKLINT/action/storage_attestation","attest_author":"https://pith.science/pith/QYYA4LTUJE5KUVGTSKOCKKLINT/action/author_attestation","sign_citation":"https://pith.science/pith/QYYA4LTUJE5KUVGTSKOCKKLINT/action/citation_signature","submit_replication":"https://pith.science/pith/QYYA4LTUJE5KUVGTSKOCKKLINT/action/replication_record"}},"created_at":"2026-05-18T02:28:21.232009+00:00","updated_at":"2026-05-18T02:28:21.232009+00:00"}