I have read that the Riemann Hypothesis is equivalent to

$\pi(x)=\text{Li}(x)+O(\sqrt{x}\log x)$

Is there an analogous statement saying the Riemann Hypothesis is equivalent to

$\pi(x)=\frac{x}{\log x}+ O(f(x))\quad$ for some $f$

or

$\pi(x)=\frac{x}{\log x}+ g(x) + O(h(x))\quad$ for some elementary function $g$ and $h$

I'm guessing that $f$ could not possibly be $\sqrt{x}\log x$ because I plotted

$\frac{\text{Li}(x)-x/\log(x)}{\sqrt x\log x}$ and it looked like it grew without bound as $x$ goes to infinity.