Point at which Prime-Counting Function becomes less than Eulerian Logarithmic Integral/Not using Riemann Hypothesis

Theorem
Let $\map \pi n$ denote the prime-counting function.

Let $a \uparrow b$ be interpreted as Knuth notation for $a^b$.

Suppose the Riemann Hypothesis does not hold.

Then:


 * $\exists n < 10 \uparrow \paren {10 \uparrow \paren {10 \uparrow 964} }: \map \pi n - \ds \int_2^n \frac {\d x} {\ln x} < 0$