Sur la variation quadratique totale de la suite des parties fractionnaires des quotients d'un nombre r\'eel positif par les nombres entiers naturels cons\'ecutifs

TL;DR

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

Contribution

It provides a novel asymptotic analysis of the quadratic total variation for fractional parts of quotients, a topic not extensively explored before.

Findings

Established an asymptotic formula for the quadratic total variation

Quantified the fluctuation behavior of fractional parts of quotients

Enhanced theoretical understanding of fractional part sequences

Abstract

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