{"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"}