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

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

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

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