On the asymptotic behavior of the integral T(x)

TL;DR

This paper analyzes the long-term behavior of the integral T(x), involving the fractional part function, providing insights into its asymptotic properties for positive real x.

Contribution

The paper offers a new estimation of the integral T(x) and explores its asymptotic behavior, which was previously not well-understood.

Findings

Derived asymptotic estimates for T(x)

Established bounds for the integral involving fractional parts

Provided insights into the growth rate of T(x) as x increases

Abstract

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