# Sign of Difference between Prime-Counting Function and Eulerian Logarithmic Integral Changes Infinitely Often

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## Theorem

The sign of $\map \pi n - \ds \int_2^n \frac {\d x} {\ln x}$ changes infinitely often as $n$ increases indefinitely.

## Proof

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## Historical Note

For values of $n$ lower than some large number, $\map \pi n - \ds \int_2^n \frac {\d x} {\ln x}$ is greater than $0$.

However, in $1914$, John Edensor Littlewood proved that at some point $\map \pi n - \ds \int_2^n \frac {\d x} {\ln x}$ switches to being less than $0$.

Furthermore, he proved that the lead changes an infinite number of times, if $n$ becomes large enough.

## Sources

- 1914: J.E. Littlewood:
*Sur la distribution des nombres premiers*(*C.R. Acad. Sci.***Vol. 158**: pp. 1869 – 1872)

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $10^{10^{10^{34}}}$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $10^{10^{10^{34}}}$