# Sur la variation de certaines suites de parties fractionnaires

Sur la variation de certaines suites de parties fractionnaires AbstractLet b > a > 0. We prove the following asymptotic formula ∑n⩾0|{x/(n+a)}-{x/(n+b)}|=2πζ(3/2)cx+O(c2/9x4/9),\sum\limits_{n \geqslant 0} {\left| {\left\{ {x/\left( {n + a} \right)} \right\} - \left\{ {x/\left( {n + b} \right)} \right\}} \right| = {2 \over \pi }\zeta \left( {3/2} \right)\sqrt {cx} + O\left( {{c^{2/9}}{x^{4/9}}} \right),} with c = b − a, uniformly for x ⩾ 40c−5(1 + b)27/2. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematics de Gruyter

# Sur la variation de certaines suites de parties fractionnaires

, Volume 29 (3): 24 – Dec 1, 2021
24 pages      /lp/de-gruyter/sur-la-variation-de-certaines-suites-de-parties-fractionnaires-CFdVaAxH0D
Publisher
de Gruyter
eISSN
2336-1298
DOI
10.2478/cm-2020-0021
Publisher site
See Article on Publisher Site

### Abstract

AbstractLet b > a > 0. We prove the following asymptotic formula ∑n⩾0|{x/(n+a)}-{x/(n+b)}|=2πζ(3/2)cx+O(c2/9x4/9),\sum\limits_{n \geqslant 0} {\left| {\left\{ {x/\left( {n + a} \right)} \right\} - \left\{ {x/\left( {n + b} \right)} \right\}} \right| = {2 \over \pi }\zeta \left( {3/2} \right)\sqrt {cx} + O\left( {{c^{2/9}}{x^{4/9}}} \right),} with c = b − a, uniformly for x ⩾ 40c−5(1 + b)27/2.

### Journal

Communications in Mathematicsde Gruyter

Published: Dec 1, 2021

Keywords: Fractional part; Elementary methods; van der Corput estimates; 11N37