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

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


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.