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Then \\[ \\sum_{n=1}^{N} R(n) =\\frac{N^{2}}{2} -2 \\sum_{\\rho} \\frac{N^{\\rho + 1}}{\\rho (\\rho + 1)} + O(N \\log^{3}N), \\] where $\\rho=1/2+i\\gamma$ runs over the non-trivial zeros of the Riemann zeta function $\\zeta(s)$."},"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":"1011.3198","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-11-14T09:03:49Z","cross_cats_sorted":[],"title_canon_sha256":"d87f8e210db856495632f3ccc8620c0fcf3c9e8150fd0526d08d62411fb55fb8","abstract_canon_sha256":"5f65f001fc4d43126b000d601563042cc7e55eef263346daf08de79febfd3a3e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:33:50.359010Z","signature_b64":"Ytwj38NvqWoLHFzQmCfEPkax+Eg8MZyyF8VaUVVqMAF1iP+hrg9tZpeN8RwxUaPnK0OeGV5zPwJr4MV09IEPAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"479cc6a39f6b2e12817f510fb7006cad77904abda1ef0595874b9bd7457507ee","last_reissued_at":"2026-05-18T03:33:50.358291Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:33:50.358291Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The number of Goldbach representations of an integer","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alessandro Languasco, Alessandro Zaccagnini","submitted_at":"2010-11-14T09:03:49Z","abstract_excerpt":"We prove the following result: Let $N \\geq 2$ and assume the Riemann Hypothesis (RH) holds. 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