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Liu (2016) improved Walfisz's error term estimate to \\[ \\sum_{n\\le x}\\varphi(n) = \\frac{3}{\\pi^2}x^2+O(x(\\log x)^{\\frac{2}{3}}(\\log\\log x)^{\\frac{1}{3}}). \\] We generalize Liu's result to a certain class of arithmetic functions and improve the result of Balakrishnan and P\\'etermann (1996). To this end, we provid"},"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":"1811.02556","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-11-06T18:57:00Z","cross_cats_sorted":[],"title_canon_sha256":"afd6cc2520fafe2b6e2aa5065de8a111277fc26acf41f55e602b53b763828d2c","abstract_canon_sha256":"45df09745c9bd91e11e1c87ef9edcc90c5968c5d4f5d080c5ff3b081f4c98abf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:30.220702Z","signature_b64":"4PoPEuAWeiVYC+s/sGfQrOrjb2TpwZRQ4q/f+HeZDEZ016fHVkJrXVpGwLtmBezivPUylxJbt9uiHQMIyVl4Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"791e67142a77445c23edcfedc2ed8f9d48924744dc1813bc6b7d2c515aeef356","last_reissued_at":"2026-05-17T23:58:30.220166Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:30.220166Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On error term estimates \\`a la Walfisz for mean values of arithmetic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Yuta Suzuki","submitted_at":"2018-11-06T18:57:00Z","abstract_excerpt":"Walfisz (1963) proved the asymptotic formula \\[ \\sum_{n\\le x}\\varphi(n) = \\frac{3}{\\pi^2}x^2+O(x(\\log x)^{\\frac{2}{3}}(\\log\\log x)^{\\frac{4}{3}}), \\] which improved the error term estimate of Mertens (1874) and had been the best possible estimate for more than 50 years. Recently, H.-Q. Liu (2016) improved Walfisz's error term estimate to \\[ \\sum_{n\\le x}\\varphi(n) = \\frac{3}{\\pi^2}x^2+O(x(\\log x)^{\\frac{2}{3}}(\\log\\log x)^{\\frac{1}{3}}). \\] We generalize Liu's result to a certain class of arithmetic functions and improve the result of Balakrishnan and P\\'etermann (1996). 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