Authors: Michel Balazard ¹; Leila Benferhat ²; Mihoub Bouderbala ²

• 1 Institut de Mathématiques de Marseille
• 2 Université des Sciences et de la Technologie Houari Boumediene = University of Sciences and Technology Houari Boumediene [Alger]

Let $b > a > 0$. We prove the following asymptotic formula :\begin{equation*}\sum_{n\ge 0} \big\lvert\{x/(n+a)\}-\{x/(n+b)\}\big\rvert=\frac{2}{\pi}\zeta(3/2)\sqrt{cx}+O(c^{2/9}x^{4/9}),\\\end{equation*}with $c=b-a$, uniformly for $x \ge 40 c^{-5}(1+b)^{27/2}$.