{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:ZVDROV4HV3TZB5UUVS25POQZMQ","short_pith_number":"pith:ZVDROV4H","schema_version":"1.0","canonical_sha256":"cd47175787aee790f694acb5d7ba19643d711bfa61181dbf8d8ab5a02d6c0daf","source":{"kind":"arxiv","id":"1206.0251","version":1},"attestation_state":"computed","paper":{"title":"A Ces\\`aro Average of Goldbach 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":"2012-06-01T17:12:10Z","abstract_excerpt":"Let $\\Lambda$ be the von Mangoldt function and $(r_G(n) = \\sum_{m_1 + m_2 = n} \\Lambda(m_1) \\Lambda(m_2))$ be the counting function for the Goldbach numbers. Let $N \\geq 2$ be an integer. We prove that $$\\begin{align} &\\sum_{n \\le N} r_G(n) \\frac{(1 - n/N)^k}{\\Gamma(k + 1)} = \\frac{N^2}{\\Gamma(k + 3)} - 2 \\sum_\\rho \\frac{\\Gamma(\\rho)}{\\Gamma(\\rho + k + 2)} N^{\\rho+1}\\\\ &\\qquad+ \\sum_{\\rho_1} \\sum_{\\rho_2} \\frac{\\Gamma(\\rho_1) \\Gamma(\\rho_2)}{\\Gamma(\\rho_1 + \\rho_2 + k + 1)} N^{\\rho_1 + \\rho_2} +  \\mathcal{O}_k(N^{1/2}), \\end{align}$$ for $k > 1$, where $\\rho$, with or without subscripts, runs "},"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":"1206.0251","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-06-01T17:12:10Z","cross_cats_sorted":[],"title_canon_sha256":"cd9aeea37fc058f9b7d09da85139790d0ff2e08c33588b06521c8fa3c6f61734","abstract_canon_sha256":"03470884702a9550ff28c506ca7a7b04154a09a70d1c1ebec07b9e39ade43be1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:46.339550Z","signature_b64":"LpkFO4sACsrrGlBJ9wQhtFash81e1KzkV+CNMhHb5QH1YZyo8/G9JByuiJ0h9PLJtYBnJlYS8dKjGyLqIZPCDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cd47175787aee790f694acb5d7ba19643d711bfa61181dbf8d8ab5a02d6c0daf","last_reissued_at":"2026-05-18T00:12:46.338809Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:46.338809Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Ces\\`aro Average of Goldbach 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":"2012-06-01T17:12:10Z","abstract_excerpt":"Let $\\Lambda$ be the von Mangoldt function and $(r_G(n) = \\sum_{m_1 + m_2 = n} \\Lambda(m_1) \\Lambda(m_2))$ be the counting function for the Goldbach numbers. Let $N \\geq 2$ be an integer. We prove that $$\\begin{align} &\\sum_{n \\le N} r_G(n) \\frac{(1 - n/N)^k}{\\Gamma(k + 1)} = \\frac{N^2}{\\Gamma(k + 3)} - 2 \\sum_\\rho \\frac{\\Gamma(\\rho)}{\\Gamma(\\rho + k + 2)} N^{\\rho+1}\\\\ &\\qquad+ \\sum_{\\rho_1} \\sum_{\\rho_2} \\frac{\\Gamma(\\rho_1) \\Gamma(\\rho_2)}{\\Gamma(\\rho_1 + \\rho_2 + k + 1)} N^{\\rho_1 + \\rho_2} +  \\mathcal{O}_k(N^{1/2}), \\end{align}$$ for $k > 1$, where $\\rho$, with or without subscripts, runs "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.0251","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":"1206.0251","created_at":"2026-05-18T00:12:46.338931+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.0251v1","created_at":"2026-05-18T00:12:46.338931+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.0251","created_at":"2026-05-18T00:12:46.338931+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZVDROV4HV3TZ","created_at":"2026-05-18T12:27:30.460161+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZVDROV4HV3TZB5UU","created_at":"2026-05-18T12:27:30.460161+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZVDROV4H","created_at":"2026-05-18T12:27:30.460161+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/ZVDROV4HV3TZB5UUVS25POQZMQ","json":"https://pith.science/pith/ZVDROV4HV3TZB5UUVS25POQZMQ.json","graph_json":"https://pith.science/api/pith-number/ZVDROV4HV3TZB5UUVS25POQZMQ/graph.json","events_json":"https://pith.science/api/pith-number/ZVDROV4HV3TZB5UUVS25POQZMQ/events.json","paper":"https://pith.science/paper/ZVDROV4H"},"agent_actions":{"view_html":"https://pith.science/pith/ZVDROV4HV3TZB5UUVS25POQZMQ","download_json":"https://pith.science/pith/ZVDROV4HV3TZB5UUVS25POQZMQ.json","view_paper":"https://pith.science/paper/ZVDROV4H","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.0251&json=true","fetch_graph":"https://pith.science/api/pith-number/ZVDROV4HV3TZB5UUVS25POQZMQ/graph.json","fetch_events":"https://pith.science/api/pith-number/ZVDROV4HV3TZB5UUVS25POQZMQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZVDROV4HV3TZB5UUVS25POQZMQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZVDROV4HV3TZB5UUVS25POQZMQ/action/storage_attestation","attest_author":"https://pith.science/pith/ZVDROV4HV3TZB5UUVS25POQZMQ/action/author_attestation","sign_citation":"https://pith.science/pith/ZVDROV4HV3TZB5UUVS25POQZMQ/action/citation_signature","submit_replication":"https://pith.science/pith/ZVDROV4HV3TZB5UUVS25POQZMQ/action/replication_record"}},"created_at":"2026-05-18T00:12:46.338931+00:00","updated_at":"2026-05-18T00:12:46.338931+00:00"}