{"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"}