On the asymptotic behaviour of the integral $T(x) =\int_{0}^{+\infty} \left|\left\{\frac{x}{t}\right\} -\left\{\frac{x}{t+1}\right\}\right|\dt$

Abstract

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