Point at which Prime-Counting Function becomes less than Eulerian Logarithmic Integral
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Theorem
Let $\map \pi n$ denote the prime-counting function.
Let $a \uparrow b$ be interpreted as Knuth notation for $a^b$.
Using Riemann Hypothesis
Suppose the Riemann Hypothesis holds.
Then:
- $\exists n < 10 \uparrow \paren {10 \uparrow \paren {10 \uparrow 34} }: \map \pi n - \ds \int_2^n \frac {\d x} {\ln x} < 0$
Not using Riemann Hypothesis
Suppose the Riemann Hypothesis does not hold.
Then:
- $\exists n < 10 \uparrow \paren {10 \uparrow \paren {10 \uparrow 964} }: \map \pi n - \ds \int_2^n \frac {\d x} {\ln x} < 0$