Sur la variation de certaines suites de parties fractionnaires

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