Sur la variation de certaines suites de parties fractionnaires

Abstract

Let $b>a>0$. We prove the following asymptotic formula $$\sum _{n\ge 0}|\{x/(n+a)\}-\{x/(n+b)\}|=\frac{2}{\pi }\zeta (3/2)\sqrt{cx}+O(c^{2/9}{x}^{4/9}),$$ with $c=b-a$, uniformly for $x\ge 40c^{-5}(1+b{)}^{27/2}$.

in French. Communications in Mathematics, University of Ostrava, A para{î}tre