Sur la variation totale de la suite des parties fractionnaires des quotients d'un nombre r{\'e}el positif par les nombres entiers naturels cons{\'e}cutifs

TL;DR

This paper derives an asymptotic formula for the total variation of the sequence of fractional parts of a positive real number divided by consecutive natural numbers, advancing understanding of fractional part behavior.

Contribution

It provides a new asymptotic formula for the total variation of fractional parts of quotients, a novel result in the analysis of such sequences.

Findings

Asymptotic formula for total variation derived

Improved understanding of fractional part sequences

Potential applications in number theory and analysis

Abstract

We give an asymptotic formula for the total variation of the sequence of fractional parts of the quotients of a positive real number by the consecutive natural numbers.