$$\Delta (x)=\left(\pi (x)-\operatorname {R} (x)+{\frac {1}{\ln x}}-{\frac {1}{\pi }}\arctan {\frac {\pi }{\ln x}}\right){\frac {\ln x}{\sqrt {x}}},$$ where $\operatorname{R}(x)$ is Riemann R function.
This page furtherly talks about $\Delta(x)$ and shows a plot of it:
And it seems true that $|\Delta(x)|\le c$ for some constant $c$. However, I searched for that and failed. Is that true ?