{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:FUZ35CW6FG5QGEY7DIBMN6BDNL","short_pith_number":"pith:FUZ35CW6","schema_version":"1.0","canonical_sha256":"2d33be8ade29bb03131f1a02c6f8236ac60fab123aa309837879e02fb7bff25c","source":{"kind":"arxiv","id":"1803.03291","version":1},"attestation_state":"computed","paper":{"title":"Rapidly converging formulae for $\\zeta(4k\\pm 1)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Blake Wilkerson, Shubho Banerjee","submitted_at":"2018-03-09T18:50:20Z","abstract_excerpt":"We provide rapidly converging formulae for the Riemann zeta function at odd integers using the Lambert series $\\mathscr{L}_q(s) = \\sum_{n=1}^\\infty n^{s} q^{n}/(1-q^n)$, $s=-(4k\\pm 1)$. Our main formula for $\\zeta(4k-1)$ converges at rate of about $e^{-\\sqrt{15}\\pi}$ per term, and the formula for $\\zeta(4k+1)$, at the rate of $e^{-4\\pi}$ per term. For example, the first order approximation yields $\\zeta(3)\\approx\\frac{\\pi ^3 \\sqrt{15}}{100} +e^{-\\sqrt{15} \\pi }\\left[\\frac{9}{4}+\\frac{4}{\\sqrt{15}}\\sinh (\\frac{\\sqrt{15} \\pi }{2})\\right]$ which has an error only of order $10^{-10}$."},"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":"1803.03291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-03-09T18:50:20Z","cross_cats_sorted":[],"title_canon_sha256":"705055257553207d8bc77ce3bf5cf703ede6cd6bc6c86de3d1f347b01e8e7a13","abstract_canon_sha256":"afedc741619961b3f240ceac4a04c2766fcb4bbe593d08bdf5f832e994ded036"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:40.094208Z","signature_b64":"dA75nIUIMmagu8psLfDgWj8uQ2gNBxzaoCT4PzZ1HTm0k2vc+kbBpWMw3m4u7sdN7fgS6rQO/4hGSfAIzHpuAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2d33be8ade29bb03131f1a02c6f8236ac60fab123aa309837879e02fb7bff25c","last_reissued_at":"2026-05-18T00:21:40.093590Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:40.093590Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rapidly converging formulae for $\\zeta(4k\\pm 1)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Blake Wilkerson, Shubho Banerjee","submitted_at":"2018-03-09T18:50:20Z","abstract_excerpt":"We provide rapidly converging formulae for the Riemann zeta function at odd integers using the Lambert series $\\mathscr{L}_q(s) = \\sum_{n=1}^\\infty n^{s} q^{n}/(1-q^n)$, $s=-(4k\\pm 1)$. Our main formula for $\\zeta(4k-1)$ converges at rate of about $e^{-\\sqrt{15}\\pi}$ per term, and the formula for $\\zeta(4k+1)$, at the rate of $e^{-4\\pi}$ per term. For example, the first order approximation yields $\\zeta(3)\\approx\\frac{\\pi ^3 \\sqrt{15}}{100} +e^{-\\sqrt{15} \\pi }\\left[\\frac{9}{4}+\\frac{4}{\\sqrt{15}}\\sinh (\\frac{\\sqrt{15} \\pi }{2})\\right]$ which has an error only of order $10^{-10}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.03291","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":"1803.03291","created_at":"2026-05-18T00:21:40.093690+00:00"},{"alias_kind":"arxiv_version","alias_value":"1803.03291v1","created_at":"2026-05-18T00:21:40.093690+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.03291","created_at":"2026-05-18T00:21:40.093690+00:00"},{"alias_kind":"pith_short_12","alias_value":"FUZ35CW6FG5Q","created_at":"2026-05-18T12:32:25.280505+00:00"},{"alias_kind":"pith_short_16","alias_value":"FUZ35CW6FG5QGEY7","created_at":"2026-05-18T12:32:25.280505+00:00"},{"alias_kind":"pith_short_8","alias_value":"FUZ35CW6","created_at":"2026-05-18T12:32:25.280505+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/FUZ35CW6FG5QGEY7DIBMN6BDNL","json":"https://pith.science/pith/FUZ35CW6FG5QGEY7DIBMN6BDNL.json","graph_json":"https://pith.science/api/pith-number/FUZ35CW6FG5QGEY7DIBMN6BDNL/graph.json","events_json":"https://pith.science/api/pith-number/FUZ35CW6FG5QGEY7DIBMN6BDNL/events.json","paper":"https://pith.science/paper/FUZ35CW6"},"agent_actions":{"view_html":"https://pith.science/pith/FUZ35CW6FG5QGEY7DIBMN6BDNL","download_json":"https://pith.science/pith/FUZ35CW6FG5QGEY7DIBMN6BDNL.json","view_paper":"https://pith.science/paper/FUZ35CW6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1803.03291&json=true","fetch_graph":"https://pith.science/api/pith-number/FUZ35CW6FG5QGEY7DIBMN6BDNL/graph.json","fetch_events":"https://pith.science/api/pith-number/FUZ35CW6FG5QGEY7DIBMN6BDNL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FUZ35CW6FG5QGEY7DIBMN6BDNL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FUZ35CW6FG5QGEY7DIBMN6BDNL/action/storage_attestation","attest_author":"https://pith.science/pith/FUZ35CW6FG5QGEY7DIBMN6BDNL/action/author_attestation","sign_citation":"https://pith.science/pith/FUZ35CW6FG5QGEY7DIBMN6BDNL/action/citation_signature","submit_replication":"https://pith.science/pith/FUZ35CW6FG5QGEY7DIBMN6BDNL/action/replication_record"}},"created_at":"2026-05-18T00:21:40.093690+00:00","updated_at":"2026-05-18T00:21:40.093690+00:00"}