Sur la variation de certaines suites de parties fractionnaires

Abstract

Let 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.