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We obtain sharp estimates of $F_{n,\\s}(f)$, for all $0< \\s\\le1$. We prove similar results for the corresponding Riemann quadratic sums $$ S_{n,\\s}(f) \\ =\\ \\sum_{1\\le k\\le \\ell \\le n}\\frac{1}{(k\\ell)^{\\s }}\\, f\\big( \\fr","authors_text":"Michel Weber","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-06-18T15:03:14Z","title":"On the Uniform Distribution (mod 1) of the Farey Sequence, quadratic Farey and Riemann sums with a remark on local integrals of $\\zeta(s)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.07628","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d20e87dad04c8ea8008b7a34448c14a04d9427f2b80eb62133bf9aa4f471ab35","target":"record","created_at":"2026-05-17T23:42:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e20e696502934389f21848eb6a79406fc7a1f1597ad3cf42650cbad259cfaa57","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-06-18T15:03:14Z","title_canon_sha256":"75c301a2d19181d02ff54723b3e64f38abb0d0d1cf96dacea10a3724eec1fb4c"},"schema_version":"1.0","source":{"id":"1906.07628","kind":"arxiv","version":1}},"canonical_sha256":"fddb368d50565a139dc32c892b4fab2d4bf65ad5c9c9943cbaab9e2c5e72aeee","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fddb368d50565a139dc32c892b4fab2d4bf65ad5c9c9943cbaab9e2c5e72aeee","first_computed_at":"2026-05-17T23:42:59.680362Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:59.680362Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CxDyb5u6sDzzvEqAPqF2aK5F5PNzT1fJhJvwzX+jlcIPv/vDILk5+fq8Uo2NjR7DNwCijftTcviboVaiSynJDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:59.680912Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.07628","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d20e87dad04c8ea8008b7a34448c14a04d9427f2b80eb62133bf9aa4f471ab35","sha256:2e915300e7adb12a26109c95fcc5ad48d0a4ea1b6460a800768284bc0acf438d"],"state_sha256":"3997dc060d42d1f1853df7f5affd53d888405a15e5b9c0b384473b4f4556d40d"}