Point at which Prime-Counting Function becomes less than Eulerian Logarithmic Integral/Using Riemann Hypothesis
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Theorem
Let $\map \pi n$ denote the prime-counting function.
Let $a \uparrow b$ be interpreted as Knuth notation for $a^b$.
Suppose the Riemann Hypothesis holds.
Then:
- $\exists n < 10 \uparrow \paren {10 \uparrow \paren {10 \uparrow 34} }: \map \pi n - \ds \int_2^n \frac {\d x} {\ln x} < 0$
Here, $10 \uparrow \paren {10 \uparrow \paren {10 \uparrow 34} }$ is Skewes' number.
Proof
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Sources
- 1933: S. Skewes: On the difference $\map \pi x − \map \Li x$ (I) (J. London Math. Soc. Vol. 8, no. 4: pp. 277 – 283)
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $10^{10^{10^{34}}}$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $10^{10^{10^{34}}}$