{"paper":{"title":"Mean Value from Representation of Rational Number as Sum of Two Egyptian Fractions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chaohua Jia","submitted_at":"2011-09-05T12:02:55Z","abstract_excerpt":"For given positive integers $n$ and $a$, let $R(n;\\,a)$ denote the number of positive integer solutions $(x,\\,y)$ of the Diophantine equation $$ {a\\over n}={1\\over x}+{1\\over y}. $$ Write $$ S(N;\\,a)=\\sum_{\\substack{n\\leq N (n,\\,a)=1}}R(n;\\,a). $$ Recently Jingjing Huang and R. C. Vaughan proved that for $4\\leq N$ and $a\\leq 2N$, there is an asymptotic formula $$ S(N;\\,a)={3\\over \\pi^2a}\\prod_{p|a}{p-1\\over p+1}\\cdot N(\\log^2N+c_1(a) \\log N+c_0(a))+\\Delta(N;\\,a). $$ In this paper, we shall get a more explicit expression with better error term for $c_0(a)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.0867","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"}