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There has been much interest and progress in further bounding $M(x)$ under the assumption of the Riemann Hypothesis. In 2009, Soundararajan established the current best bound of \\[ M(x)\\ll\\sqrt{x}\\exp\\left((\\log x)^{1/2}(\\log\\log x)^c\\right) \\] (setting $c$ to $14$, though this can be reduced). Halupczok and Suger recently applied Soundararajan's method to bound more general sums of the M\\\"obi"},"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":"1406.7326","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-06-27T22:08:06Z","cross_cats_sorted":[],"title_canon_sha256":"03b0e9b754edb81998c3969750eca809e2ee1233b85ecc112b200d56c4f73962","abstract_canon_sha256":"7c73f0565f87add4bc68b74ae4315f50e96a44dfe89de32bd4a9aece07a542c9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:45.609580Z","signature_b64":"NWdOsQfcWBNJ+HW1iVP2WNx5wCtbcpbiKRi5V37qg5jNtoQdYMJ75z1PMXJXPSVz0zzGUNRH5V3fm9U3JC22Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47f7a8556fbdba94d70fc295a73a7127d892083caac5145d2037512bfe77a65e","last_reissued_at":"2026-05-18T02:48:45.608876Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:45.608876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounding sums of the M\\\"obius function over arithmetic progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Lynnelle Ye","submitted_at":"2014-06-27T22:08:06Z","abstract_excerpt":"Let $M(x)=\\sum_{1\\le n\\le x}\\mu(n)$ where $\\mu$ is the M\\\"obius function. 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