{"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"}