{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:XSOS3OTDZTUHUUAHM5QZ4YURVI","short_pith_number":"pith:XSOS3OTD","schema_version":"1.0","canonical_sha256":"bc9d2dba63cce87a500767619e6291aa1a7da286da6b171e05fab7a4499c3039","source":{"kind":"arxiv","id":"1206.0255","version":1},"attestation_state":"computed","paper":{"title":"A Ces\\`aro Average of Hardy-Littlewood numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alessandro Languasco, Alessandro Zaccagnini","submitted_at":"2012-06-01T17:18:30Z","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"},"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":"1206.0255","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-06-01T17:18:30Z","cross_cats_sorted":[],"title_canon_sha256":"0439ec95beb042a3e0efae390497fe38a608e09dae2ea66c113d4bb1441a1162","abstract_canon_sha256":"cd858d46e92668437002b1fcea214b3999096709b7cc52a166e3dda2ca91e33c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:46.331635Z","signature_b64":"JX/C0yfjfI2kdj/fjSVlDAv9KT7G0J0cCS6EEJCWmG4PD0SJGeYG9yyhgCYGDGLULuxM2UipSovy4VKsuc2DBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bc9d2dba63cce87a500767619e6291aa1a7da286da6b171e05fab7a4499c3039","last_reissued_at":"2026-05-18T00:12:46.330977Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:46.330977Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Ces\\`aro Average of Hardy-Littlewood numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alessandro Languasco, Alessandro Zaccagnini","submitted_at":"2012-06-01T17:18:30Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.0255","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":"1206.0255","created_at":"2026-05-18T00:12:46.331099+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.0255v1","created_at":"2026-05-18T00:12:46.331099+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.0255","created_at":"2026-05-18T00:12:46.331099+00:00"},{"alias_kind":"pith_short_12","alias_value":"XSOS3OTDZTUH","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_16","alias_value":"XSOS3OTDZTUHUUAH","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_8","alias_value":"XSOS3OTD","created_at":"2026-05-18T12:27:27.928770+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/XSOS3OTDZTUHUUAHM5QZ4YURVI","json":"https://pith.science/pith/XSOS3OTDZTUHUUAHM5QZ4YURVI.json","graph_json":"https://pith.science/api/pith-number/XSOS3OTDZTUHUUAHM5QZ4YURVI/graph.json","events_json":"https://pith.science/api/pith-number/XSOS3OTDZTUHUUAHM5QZ4YURVI/events.json","paper":"https://pith.science/paper/XSOS3OTD"},"agent_actions":{"view_html":"https://pith.science/pith/XSOS3OTDZTUHUUAHM5QZ4YURVI","download_json":"https://pith.science/pith/XSOS3OTDZTUHUUAHM5QZ4YURVI.json","view_paper":"https://pith.science/paper/XSOS3OTD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.0255&json=true","fetch_graph":"https://pith.science/api/pith-number/XSOS3OTDZTUHUUAHM5QZ4YURVI/graph.json","fetch_events":"https://pith.science/api/pith-number/XSOS3OTDZTUHUUAHM5QZ4YURVI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XSOS3OTDZTUHUUAHM5QZ4YURVI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XSOS3OTDZTUHUUAHM5QZ4YURVI/action/storage_attestation","attest_author":"https://pith.science/pith/XSOS3OTDZTUHUUAHM5QZ4YURVI/action/author_attestation","sign_citation":"https://pith.science/pith/XSOS3OTDZTUHUUAHM5QZ4YURVI/action/citation_signature","submit_replication":"https://pith.science/pith/XSOS3OTDZTUHUUAHM5QZ4YURVI/action/replication_record"}},"created_at":"2026-05-18T00:12:46.331099+00:00","updated_at":"2026-05-18T00:12:46.331099+00:00"}