# Sur la variation de certaines suites de parties fractionnaires

## TL;DR

This paper derives an asymptotic formula for the sum of absolute differences of fractional parts of scaled sequences, revealing their asymptotic behavior with explicit error bounds.

## Contribution

It provides a new asymptotic expression for sums involving fractional parts of sequences with explicit error terms and uniform bounds.

## Key findings

- Asymptotic formula involving zeta function and square root of x
- Explicit error term with power bounds
- Uniform validity for large x and parameter ranges

## Abstract

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