{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:XP6D7SW3TGJ3QYKHMEAF6V5Q2C","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":"dfe9d43009b900dc04afa93ccf35ccb572b8966ac332ce83be0a13a033c7334b","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-09-27T19:56:59Z","title_canon_sha256":"8596bf604c35f0176905dfb4f7d0a4eb5c577ebcea7a76a321b564d84c670fb2"},"schema_version":"1.0","source":{"id":"1009.5375","kind":"arxiv","version":8}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.5375","created_at":"2026-05-18T01:29:51Z"},{"alias_kind":"arxiv_version","alias_value":"1009.5375v8","created_at":"2026-05-18T01:29:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.5375","created_at":"2026-05-18T01:29:51Z"},{"alias_kind":"pith_short_12","alias_value":"XP6D7SW3TGJ3","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XP6D7SW3TGJ3QYKH","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XP6D7SW3","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:4e0f261380821804d666a8e82a24a71475ebdae5328494ae881947174c0a0b12","target":"graph","created_at":"2026-05-18T01:29:51Z","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 $H_n^{(2)}$ denote the second-order harmonic number $\\sum_{0<k\\le n}1/k^2$ for $n=0,1,2,\\ldots$. In this paper we obtain the following identity: $$\\sum_{k=1}^\\infty\\frac{2^kH_{k-1}^{(2)}}{k\\binom{2k}k}=\\frac{\\pi^3}{48}.$$ We explain how we found the series and develop related congruences involving Bernoulli or Euler numbers; for example, it is shown that $$\\sum_{k=1}^{p-1}\\frac{\\binom{2k}k}{2^k}H_k^{(2)}\\equiv-E_{p-3}\\pmod{p}$$ for any prime $p>3$, where $E_0,E_1,E_2,\\ldots$ are Euler numbers. Motivated by the Amdeberhan-Zeilberger identity $\\sum_{k=1}^\\infty(21k-8)/(k^3\\binom{2k}k^3)=\\pi^","authors_text":"Zhi-Wei Sun","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-09-27T19:56:59Z","title":"A new series for $\\pi^3$ and related congruences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.5375","kind":"arxiv","version":8},"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:e30c26d8e0a20493239e59413761a381be03db3b1c7953ef6cf04d0e0d9e164d","target":"record","created_at":"2026-05-18T01:29:51Z","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":"dfe9d43009b900dc04afa93ccf35ccb572b8966ac332ce83be0a13a033c7334b","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-09-27T19:56:59Z","title_canon_sha256":"8596bf604c35f0176905dfb4f7d0a4eb5c577ebcea7a76a321b564d84c670fb2"},"schema_version":"1.0","source":{"id":"1009.5375","kind":"arxiv","version":8}},"canonical_sha256":"bbfc3fcadb9993b8614761005f57b0d09a4a183d99cedb5e1442b7dcb20f46d5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bbfc3fcadb9993b8614761005f57b0d09a4a183d99cedb5e1442b7dcb20f46d5","first_computed_at":"2026-05-18T01:29:51.796378Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:29:51.796378Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8oZrzLZOGOK4GDAIY1bl1aMaEopgelo91smomTyz/1eY4mMcfBnHl1z7jPAusZVv2NMENS9DMOG49OGCmOa4Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:29:51.796808Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.5375","source_kind":"arxiv","source_version":8}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e30c26d8e0a20493239e59413761a381be03db3b1c7953ef6cf04d0e0d9e164d","sha256:4e0f261380821804d666a8e82a24a71475ebdae5328494ae881947174c0a0b12"],"state_sha256":"9d04219b6775a5947cdc4cf80d6c4db83d8bb5a8f41fc4e223593a8775112360"}