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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 $c_0(a)$."},"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":"1109.0867","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-05T12:02:55Z","cross_cats_sorted":[],"title_canon_sha256":"17d6c29e39811a1f4062c90a305a948a6cfb9ebc7bcf303ec6294a89760d42ab","abstract_canon_sha256":"4f922a822f6d8ba034e411f53274372c343edd36fcc2f9caf7743c529698e804"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:06.498514Z","signature_b64":"cIBTqMAFafU+R+iMDZ/iVg5Ja2B48XmhbFUplOrln3gH0ie4LXUXYN06KWg36BuzaXUrfViPcx83gnW5OSVnAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e62a4f5104bb3f7d42c59c47990fa2e251ae937a312326eab87a28e94161aff2","last_reissued_at":"2026-05-18T04:14:06.497936Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:06.497936Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1109.0867","created_at":"2026-05-18T04:14:06.498015+00:00"},{"alias_kind":"arxiv_version","alias_value":"1109.0867v1","created_at":"2026-05-18T04:14:06.498015+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.0867","created_at":"2026-05-18T04:14:06.498015+00:00"},{"alias_kind":"pith_short_12","alias_value":"4YVE6UIEXM7X","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_16","alias_value":"4YVE6UIEXM7X2QWF","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_8","alias_value":"4YVE6UIE","created_at":"2026-05-18T12:26:20.644004+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/4YVE6UIEXM7X2QWFTRDZSD5C4J","json":"https://pith.science/pith/4YVE6UIEXM7X2QWFTRDZSD5C4J.json","graph_json":"https://pith.science/api/pith-number/4YVE6UIEXM7X2QWFTRDZSD5C4J/graph.json","events_json":"https://pith.science/api/pith-number/4YVE6UIEXM7X2QWFTRDZSD5C4J/events.json","paper":"https://pith.science/paper/4YVE6UIE"},"agent_actions":{"view_html":"https://pith.science/pith/4YVE6UIEXM7X2QWFTRDZSD5C4J","download_json":"https://pith.science/pith/4YVE6UIEXM7X2QWFTRDZSD5C4J.json","view_paper":"https://pith.science/paper/4YVE6UIE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1109.0867&json=true","fetch_graph":"https://pith.science/api/pith-number/4YVE6UIEXM7X2QWFTRDZSD5C4J/graph.json","fetch_events":"https://pith.science/api/pith-number/4YVE6UIEXM7X2QWFTRDZSD5C4J/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4YVE6UIEXM7X2QWFTRDZSD5C4J/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4YVE6UIEXM7X2QWFTRDZSD5C4J/action/storage_attestation","attest_author":"https://pith.science/pith/4YVE6UIEXM7X2QWFTRDZSD5C4J/action/author_attestation","sign_citation":"https://pith.science/pith/4YVE6UIEXM7X2QWFTRDZSD5C4J/action/citation_signature","submit_replication":"https://pith.science/pith/4YVE6UIEXM7X2QWFTRDZSD5C4J/action/replication_record"}},"created_at":"2026-05-18T04:14:06.498015+00:00","updated_at":"2026-05-18T04:14:06.498015+00:00"}