On the Asymptotic Behaviour of the Integral T(x)=∫0+∞|{xt}−{xt+1}|dtT(x)=∫_0+∞|\{x/t\}-\{x/t+1\}|\,dt

Mihoub Bouderbala; Meselem Karras

doi:10.2478/tmmp-2026-0009

June 30, 2026Tatra Mountains Mathematical PublicationsOpen Access

On the Asymptotic Behaviour of the Integral T(x)=∫0+∞|{xt}−{xt+1}|dtT(x)=∫_0+∞|\{x/t\}-\{x/t+1\}|\,dt

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Key Points

The aim is to estimate the integral T(x) defined with fractional parts for any positive real number x.
Estimation of the integral T(x) using methods from mathematical analysis.
Involves analyzing the behaviour of fractional parts as t varies over positive real numbers.
Provides specific asymptotic estimates for T(x) as x approaches infinity.
Demonstrates the integral's dependence on the behaviour of fractional parts for large values of x.

Implication

Estimates the integral T(x) involving fractional parts of real numbers, providing insights into its asymptotic behaviour.

Abstract

In this paper, we estimate the integral T ( x ) considered in the title, where {t} denote the fractional parts of the real number t ,and x is any positive real number.

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Cite This Study

Bouderbala et al. (2026) studied this question.

synapsesocial.com/papers/6a45fffc9ed1343031310395 https://doi.org/10.2478/tmmp-2026-0009

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On the Asymptotic Behaviour of the Integral T(x)=∫0+∞|{xt}−{xt+1}|dtT(x)=∫_0+∞|\{x/t\}-\{x/t+1\}|\,dt

Key Points

Implication

Abstract

Discussion

Cite This Study

On the Asymptotic Behaviour of the Integral T(x)=∫0+∞|{xt}−{xt+1}|dtT(x)=∫_0+∞|\{x/t\}-\{x/t+1\}|\,dt

Key Points

Implication

Abstract

Discussion

Cite This Study

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