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:

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

Sources

• 1955: S. Skewes: On the difference $\map \pi x − \map \Li x$ (II) (Proc. London Math. Soc. Ser. 3 Vol. 5, no. 17: pp. 48 – 70)