{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:5ETSQMVE2BWTWETRJMGSNOISGE","short_pith_number":"pith:5ETSQMVE","schema_version":"1.0","canonical_sha256":"e9272832a4d06d3b12714b0d26b9123123068bceb418c531c6cdc5d36c3c3819","source":{"kind":"arxiv","id":"1802.01473","version":6},"attestation_state":"computed","paper":{"title":"Two $q$-analogues of Euler's formula $\\zeta(2)=\\pi^2/6$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2018-02-05T15:52:27Z","abstract_excerpt":"It is well known that $\\zeta(2)=\\pi^2/6$ as discovered by Euler. In this paper we present the following two $q$-analogues of this celebrated formula: $$\\sum_{k=0}^\\infty\\frac{q^k(1+q^{2k+1})}{(1-q^{2k+1})^2}=\\prod_{n=1}^\\infty\\frac{(1-q^{2n})^4}{(1-q^{2n-1})^4}$$ and $$\\sum_{k=0}^\\infty\\frac{q^{2k-\\lfloor(-1)^kk/2\\rfloor}}{(1-q^{2k+1})^2} =\\prod_{n=1}^\\infty\\frac{(1-q^{2n})^2(1-q^{4n})^2}{(1-q^{2n-1})^2(1-q^{4n-2})^2},$$ where $q$ is any complex number with $|q|<1$. We also give a $q$-analogue of the identity $\\zeta(4)=\\pi^4/90$, and pose a problem on $q$-analogues of Euler's formula for $\\zet"},"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":"1802.01473","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-05T15:52:27Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a913e266bb581eb47abc3b5f42edee06543a9529e2597ae3292126c523345824","abstract_canon_sha256":"13d66ee412cf60934666199e3bb3ccfb04d0f7dbb87cdc003e2f8813da3e7118"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:20:09.935946Z","signature_b64":"GU2AhP1f/ZrhE93JfInNSwaV21s75dCFZvIURG3DQ31ahv3F/h13Q2AG1efaG3O+n4xr+qIHZUSCyBqemC/0AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e9272832a4d06d3b12714b0d26b9123123068bceb418c531c6cdc5d36c3c3819","last_reissued_at":"2026-07-05T00:20:09.935538Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:20:09.935538Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two $q$-analogues of Euler's formula $\\zeta(2)=\\pi^2/6$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2018-02-05T15:52:27Z","abstract_excerpt":"It is well known that $\\zeta(2)=\\pi^2/6$ as discovered by Euler. In this paper we present the following two $q$-analogues of this celebrated formula: $$\\sum_{k=0}^\\infty\\frac{q^k(1+q^{2k+1})}{(1-q^{2k+1})^2}=\\prod_{n=1}^\\infty\\frac{(1-q^{2n})^4}{(1-q^{2n-1})^4}$$ and $$\\sum_{k=0}^\\infty\\frac{q^{2k-\\lfloor(-1)^kk/2\\rfloor}}{(1-q^{2k+1})^2} =\\prod_{n=1}^\\infty\\frac{(1-q^{2n})^2(1-q^{4n})^2}{(1-q^{2n-1})^2(1-q^{4n-2})^2},$$ where $q$ is any complex number with $|q|<1$. We also give a $q$-analogue of the identity $\\zeta(4)=\\pi^4/90$, and pose a problem on $q$-analogues of Euler's formula for $\\zet"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01473","kind":"arxiv","version":6},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1802.01473/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1802.01473","created_at":"2026-07-05T00:20:09.935591+00:00"},{"alias_kind":"arxiv_version","alias_value":"1802.01473v6","created_at":"2026-07-05T00:20:09.935591+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.01473","created_at":"2026-07-05T00:20:09.935591+00:00"},{"alias_kind":"pith_short_12","alias_value":"5ETSQMVE2BWT","created_at":"2026-07-05T00:20:09.935591+00:00"},{"alias_kind":"pith_short_16","alias_value":"5ETSQMVE2BWTWETR","created_at":"2026-07-05T00:20:09.935591+00:00"},{"alias_kind":"pith_short_8","alias_value":"5ETSQMVE","created_at":"2026-07-05T00:20:09.935591+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/5ETSQMVE2BWTWETRJMGSNOISGE","json":"https://pith.science/pith/5ETSQMVE2BWTWETRJMGSNOISGE.json","graph_json":"https://pith.science/api/pith-number/5ETSQMVE2BWTWETRJMGSNOISGE/graph.json","events_json":"https://pith.science/api/pith-number/5ETSQMVE2BWTWETRJMGSNOISGE/events.json","paper":"https://pith.science/paper/5ETSQMVE"},"agent_actions":{"view_html":"https://pith.science/pith/5ETSQMVE2BWTWETRJMGSNOISGE","download_json":"https://pith.science/pith/5ETSQMVE2BWTWETRJMGSNOISGE.json","view_paper":"https://pith.science/paper/5ETSQMVE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1802.01473&json=true","fetch_graph":"https://pith.science/api/pith-number/5ETSQMVE2BWTWETRJMGSNOISGE/graph.json","fetch_events":"https://pith.science/api/pith-number/5ETSQMVE2BWTWETRJMGSNOISGE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5ETSQMVE2BWTWETRJMGSNOISGE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5ETSQMVE2BWTWETRJMGSNOISGE/action/storage_attestation","attest_author":"https://pith.science/pith/5ETSQMVE2BWTWETRJMGSNOISGE/action/author_attestation","sign_citation":"https://pith.science/pith/5ETSQMVE2BWTWETRJMGSNOISGE/action/citation_signature","submit_replication":"https://pith.science/pith/5ETSQMVE2BWTWETRJMGSNOISGE/action/replication_record"}},"created_at":"2026-07-05T00:20:09.935591+00:00","updated_at":"2026-07-05T00:20:09.935591+00:00"}