On the Asymptotic Behaviour of the Integral T(x)=∫0+∞|{xt}−{xt+1}|dt 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 ) considered in the title, where {t} denote the fractional parts of the real number t ,and x is any positive real number.