{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:7XNTNDKQKZNBHHODFSESWT5LFV","short_pith_number":"pith:7XNTNDKQ","canonical_record":{"source":{"id":"1906.07628","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-06-18T15:03:14Z","cross_cats_sorted":[],"title_canon_sha256":"75c301a2d19181d02ff54723b3e64f38abb0d0d1cf96dacea10a3724eec1fb4c","abstract_canon_sha256":"e20e696502934389f21848eb6a79406fc7a1f1597ad3cf42650cbad259cfaa57"},"schema_version":"1.0"},"canonical_sha256":"fddb368d50565a139dc32c892b4fab2d4bf65ad5c9c9943cbaab9e2c5e72aeee","source":{"kind":"arxiv","id":"1906.07628","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.07628","created_at":"2026-05-17T23:42:59Z"},{"alias_kind":"arxiv_version","alias_value":"1906.07628v1","created_at":"2026-05-17T23:42:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.07628","created_at":"2026-05-17T23:42:59Z"},{"alias_kind":"pith_short_12","alias_value":"7XNTNDKQKZNB","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_16","alias_value":"7XNTNDKQKZNBHHOD","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_8","alias_value":"7XNTNDKQ","created_at":"2026-05-18T12:33:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:7XNTNDKQKZNBHHODFSESWT5LFV","target":"record","payload":{"canonical_record":{"source":{"id":"1906.07628","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-06-18T15:03:14Z","cross_cats_sorted":[],"title_canon_sha256":"75c301a2d19181d02ff54723b3e64f38abb0d0d1cf96dacea10a3724eec1fb4c","abstract_canon_sha256":"e20e696502934389f21848eb6a79406fc7a1f1597ad3cf42650cbad259cfaa57"},"schema_version":"1.0"},"canonical_sha256":"fddb368d50565a139dc32c892b4fab2d4bf65ad5c9c9943cbaab9e2c5e72aeee","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:59.680912Z","signature_b64":"CxDyb5u6sDzzvEqAPqF2aK5F5PNzT1fJhJvwzX+jlcIPv/vDILk5+fq8Uo2NjR7DNwCijftTcviboVaiSynJDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fddb368d50565a139dc32c892b4fab2d4bf65ad5c9c9943cbaab9e2c5e72aeee","last_reissued_at":"2026-05-17T23:42:59.680362Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:59.680362Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1906.07628","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:42:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g2IyPhmz/pXFcDrES3hGSqVXLEhM202Eq9uDQQleFMbN4pEb+5W7e1+06WAJGSgzmd6aykK0Bt1s2aaMD2OqCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T00:47:29.876035Z"},"content_sha256":"d20e87dad04c8ea8008b7a34448c14a04d9427f2b80eb62133bf9aa4f471ab35","schema_version":"1.0","event_id":"sha256:d20e87dad04c8ea8008b7a34448c14a04d9427f2b80eb62133bf9aa4f471ab35"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:7XNTNDKQKZNBHHODFSESWT5LFV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Uniform Distribution (mod 1) of the Farey Sequence, quadratic Farey and Riemann sums with a remark on local integrals of $\\zeta(s)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michel Weber","submitted_at":"2019-06-18T15:03:14Z","abstract_excerpt":"For $1$-periodic functions $f$ satisfying only a weak local regularity assumption of Dini's type at rational points of $]0,1[$, we study the Farey sums\n  $$F_n(f)= \\sum_{\\frac{\\k}{\\l}\\in \\F_n} f\\big(\\frac{\\k}{\\l}\\big),\\qq F_{n,\\s}(f)= \\sum_{\\frac{\\k}{\\l}\\in \\F_n} \\frac{1}{\\k^\\s\\l^\\s}f\\big(\\frac{\\k}{\\l}\\big),\\qq 1/2\\le \\s<1 , $$ where $\\F_n$ is the Farey series of order $n\\ge 1$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.07628","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:42:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YyGEUWqH4eeRL4HgtmQB7r2jB8cZFRIM+Ip5ygPr9lJgv+biIK1/CkQE2VOv2CILUElAYoVqKh/Fe3aAYEzmBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T00:47:29.876514Z"},"content_sha256":"2e915300e7adb12a26109c95fcc5ad48d0a4ea1b6460a800768284bc0acf438d","schema_version":"1.0","event_id":"sha256:2e915300e7adb12a26109c95fcc5ad48d0a4ea1b6460a800768284bc0acf438d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7XNTNDKQKZNBHHODFSESWT5LFV/bundle.json","state_url":"https://pith.science/pith/7XNTNDKQKZNBHHODFSESWT5LFV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7XNTNDKQKZNBHHODFSESWT5LFV/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-23T00:47:29Z","links":{"resolver":"https://pith.science/pith/7XNTNDKQKZNBHHODFSESWT5LFV","bundle":"https://pith.science/pith/7XNTNDKQKZNBHHODFSESWT5LFV/bundle.json","state":"https://pith.science/pith/7XNTNDKQKZNBHHODFSESWT5LFV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7XNTNDKQKZNBHHODFSESWT5LFV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:7XNTNDKQKZNBHHODFSESWT5LFV","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"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}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.07628","created_at":"2026-05-17T23:42:59Z"},{"alias_kind":"arxiv_version","alias_value":"1906.07628v1","created_at":"2026-05-17T23:42:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.07628","created_at":"2026-05-17T23:42:59Z"},{"alias_kind":"pith_short_12","alias_value":"7XNTNDKQKZNB","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_16","alias_value":"7XNTNDKQKZNBHHOD","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_8","alias_value":"7XNTNDKQ","created_at":"2026-05-18T12:33:12Z"}],"graph_snapshots":[{"event_id":"sha256:2e915300e7adb12a26109c95fcc5ad48d0a4ea1b6460a800768284bc0acf438d","target":"graph","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":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"For $1$-periodic functions $f$ satisfying only a weak local regularity assumption of Dini's type at rational points of $]0,1[$, we study the Farey sums\n  $$F_n(f)= \\sum_{\\frac{\\k}{\\l}\\in \\F_n} f\\big(\\frac{\\k}{\\l}\\big),\\qq F_{n,\\s}(f)= \\sum_{\\frac{\\k}{\\l}\\in \\F_n} \\frac{1}{\\k^\\s\\l^\\s}f\\big(\\frac{\\k}{\\l}\\big),\\qq 1/2\\le \\s<1 , $$ where $\\F_n$ is the Farey series of order $n\\ge 1$. We obtain sharp estimates of $F_{n,\\s}(f)$, for all $0< \\s\\le1$. We prove similar results for the corresponding Riemann quadratic sums $$ 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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3dW7NTHzN77hdddC4/Gc9s4pCi2q7Sf6bsDIrl5BNRv1v+9L+DMu5suYEIrRA+2o4CYE4CGh7QIMnsWc9byGDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T00:47:29.879218Z","bundle_sha256":"0989c8f8e63b652f8a0e1fcfc23d2ef927b9b86b0fca98bb965e6fc77fd92af2"}}