{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:XSOS3OTDZTUHUUAHM5QZ4YURVI","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":"cd858d46e92668437002b1fcea214b3999096709b7cc52a166e3dda2ca91e33c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-06-01T17:18:30Z","title_canon_sha256":"0439ec95beb042a3e0efae390497fe38a608e09dae2ea66c113d4bb1441a1162"},"schema_version":"1.0","source":{"id":"1206.0255","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.0255","created_at":"2026-05-18T00:12:46Z"},{"alias_kind":"arxiv_version","alias_value":"1206.0255v1","created_at":"2026-05-18T00:12:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.0255","created_at":"2026-05-18T00:12:46Z"},{"alias_kind":"pith_short_12","alias_value":"XSOS3OTDZTUH","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XSOS3OTDZTUHUUAH","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XSOS3OTD","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:3d7f947900d747b89d7d04ae35627fc5f12c0ccc669898513499d5309456d6d5","target":"graph","created_at":"2026-05-18T00:12:46Z","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 $\\Lambda$ be the von Mangoldt function and $r_{\\textit{HL}}(n) = \\sum_{m_1 + m_2^2 = n} \\Lambda(m_1),$ be the counting function for the Hardy-Littlewood numbers. Let $N$ be a sufficiently large integer. We prove that $$\\begin{align}\\sum_{n \\le N} r_{\\textit{HL}}(n) \\frac{(1 - n/N)^k}{\\Gamma(k + 1)} &= \\frac{\\pi^{1 / 2}}2 \\frac{N^{3 / 2}}{\\Gamma(k + 5 / 2)} - \\frac 12 \\frac{N}{\\Gamma(k + 2)} - \\frac{\\pi^{1 / 2}}2 \\sum_{\\rho} \\frac{\\Gamma(\\rho)}{\\Gamma(k + 3 / 2 + \\rho)} N^{1 / 2 + \\rho}\\\\ &+ 1/2 \\sum_{\\rho} \\frac{\\Gamma(\\rho)}{\\Gamma(k + 1 + \\rho)} N^{\\rho} + \\frac{N^{3 / 4 - k / 2}}{\\pi^{k","authors_text":"Alessandro Languasco, Alessandro Zaccagnini","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-06-01T17:18:30Z","title":"A Ces\\`aro Average of Hardy-Littlewood numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.0255","kind":"arxiv","version":1},"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":"sha256:6340695f7746e23a4105d39dc49f4cb5e4da12f8ecbef7deca62e9c6ca1844f1","target":"record","created_at":"2026-05-18T00:12:46Z","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":"cd858d46e92668437002b1fcea214b3999096709b7cc52a166e3dda2ca91e33c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-06-01T17:18:30Z","title_canon_sha256":"0439ec95beb042a3e0efae390497fe38a608e09dae2ea66c113d4bb1441a1162"},"schema_version":"1.0","source":{"id":"1206.0255","kind":"arxiv","version":1}},"canonical_sha256":"bc9d2dba63cce87a500767619e6291aa1a7da286da6b171e05fab7a4499c3039","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bc9d2dba63cce87a500767619e6291aa1a7da286da6b171e05fab7a4499c3039","first_computed_at":"2026-05-18T00:12:46.330977Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:12:46.330977Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JX/C0yfjfI2kdj/fjSVlDAv9KT7G0J0cCS6EEJCWmG4PD0SJGeYG9yyhgCYGDGLULuxM2UipSovy4VKsuc2DBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:12:46.331635Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.0255","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6340695f7746e23a4105d39dc49f4cb5e4da12f8ecbef7deca62e9c6ca1844f1","sha256:3d7f947900d747b89d7d04ae35627fc5f12c0ccc669898513499d5309456d6d5"],"state_sha256":"ddd3175261d1a86a036b6403759a09e4567bcb436f02e8b6ee48b6ecad6d064d"}