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He also stated that, assuming the Riemann Hypothesis(RH), $$ E(x)=O(x^{{1\\over 2}+\\epsilon}). $$\n  In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $$ E(x)=O(x^{1\\over 2}(\\log x)^5\\log\\log x). $$ In 2003, K. Ramachandra and A. Sankaranarayanan proved the above re"},"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":"1311.4041","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-16T10:00:26Z","cross_cats_sorted":[],"title_canon_sha256":"ee00048333f6b39e09027b09cd9deae71fd33e9a0c1a99dd64c96120b64f80af","abstract_canon_sha256":"e55d5699669fbfba0bea4a37275dfb95741ebb33ef0971e985140376d3813539"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:49.698480Z","signature_b64":"cXcDmhrlVhMLL2hzJQuDRdzcZht6eMw/YHjNs3u2JskXgcGU2AdgYkzgsN2kxBPpyWdKKbJSSGm+dJMEVz0JAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"160af456a0e39258691864f7cdf8aecab8045078a769f34eca64e1c2bc985d17","last_reissued_at":"2026-05-18T02:55:49.697955Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:49.697955Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Mean Square of Divisor Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ayyadurai Sankaranarayanan, Chaohua Jia","submitted_at":"2013-11-16T10:00:26Z","abstract_excerpt":"Let $d(n)$ be the divisor function. In 1916, S. Ramanujan stated but without proof that $$\\sum_{n\\leq x}d^2(n)=xP(\\log x)+E(x), $$ where $P(y)$ is a cubic polynomial in $y$ and $$ E(x)=O(x^{{3\\over 5}+\\epsilon}), $$ where $\\epsilon$ is a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis(RH), $$ E(x)=O(x^{{1\\over 2}+\\epsilon}). $$\n  In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $$ E(x)=O(x^{1\\over 2}(\\log x)^5\\log\\log x). $$ In 2003, K. Ramachandra and A. 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