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We prove a couple of mean value theorems for the second moment $(R(n;a))^2$ and from which we deduce $\\log R(n;a)$ satisfies a certain Gaussian distribution with mean $\\log 3\\log\\log n$ and variance $(log 3)^2\\log\\log n$, which is an analog of the classical theorem of Erd\\H os and Kac. And finally these results in all suggest that the behavior of $R(n;a)$ resembles the divisor function $d(n^2)$ in various aspects."},"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":"1109.2274","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-11T01:33:03Z","cross_cats_sorted":[],"title_canon_sha256":"ddcfd5990590e06f0d34a97253a073d1a04ad8d46a869c8a54ccdb5a6d9926f2","abstract_canon_sha256":"9b112682a7f80cb7611d9d39c49962a7de90c6da2b62c316877c42cda66e953d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:13:38.146086Z","signature_b64":"qoqY2WrhqdkRHWPTgdsF8oU0kzxnZZVmM0nONL4g8WA5xoBPty9QFEJrVL6+3SQMTwopd34fSlBWK6LtClAxAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d7dd779005caf1d3295ab307713fd2513e696b7c25fac21d9f999489e09337dc","last_reissued_at":"2026-05-18T04:13:38.145376Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:13:38.145376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mean value theorems for binary Egyptian fractions II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jing-Jing Huang, Robert C. 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