On the Uniform Distribution (mod 1) of the Farey Sequence, quadratic Farey and Riemann sums with a remark on local integrals of $\zeta(s)$

Abstract

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 $$F_n(f)= \sum_{\frac{\k}{\l}\in \F_n} f\big(\frac{\k}{\l}\big),\qq F_{n,\s}(f)= \sum_{\frac{\k}{\l}\in \F_n} \frac{1}{\k^\s\l^\s}f\big(\frac{\k}{\l}\big),\qq 1/2\le \s<1 , $$ 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 $$ S_{n,\s}(f) \ =\ \sum_{1\le k\le \ell \le n}\frac{1}{(k\ell)^{\s }}\, f\big( \frac{k}{\ell}\big). $$ These sums are related to local integrals of the Riemann zeta-function over bounded intervals $I$, which are considered in the last part of the paper.