Sur la variation de certaines suites de parties fractionnairesArticle

• 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 $x \ge 40 c^{-5}(1+b)^{27/2}$.

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• 11N37 - Asymptotic results on arithmetic functions