Prime-Counting Function in terms of Eulerian Logarithmic Integral/Riemann Hypothesis Holds

Theorem
Let $\map \pi x$ denote the prime-counting function of a number $x$.

Let $\map \Li x$ denote the Eulerian logarithmic integral of $x$:
 * $\map \Li x := \displaystyle \int_2^x \dfrac {\d t} {\ln t}$

If the Riemann Hypothesis holds, then:
 * $\map \pi x = \map \Li x + \map {\mathcal O} {\sqrt x \, \ln x}$

where $\mathcal O$ is the big-O notation.