Sign of Difference between Prime-Counting Function and Eulerian Logarithmic Integral Changes Infinitely Often
Jump to navigation
Jump to search
Theorem
The sign of $\map \pi n - \ds \int_2^n \frac {\d x} {\ln x}$ changes infinitely often as $n$ increases indefinitely.
Proof
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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): $10^{10^{10^{34}}}$