Difference between revisions of "Logarithmic integral"
From specialfunctionswiki
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File:Logarithmicintegral.png|Graph of $\mathrm{li}$ on $[0,6]$. | File:Logarithmicintegral.png|Graph of $\mathrm{li}$ on $[0,6]$. | ||
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Revision as of 06:44, 5 April 2015
The logarithmic integral is $$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{dt}{\log(t)},$$ where $\log$ denotes the logarithm. The logarithmic integral is related to the exponential integral by the formula $$\mathrm{li}(x)=\mathrm{Ei}( \log(x)).$$
- Logarithmicintegral.png
Graph of $\mathrm{li}$ on $[0,6]$.
Contents
Properties
Theorem
The following formula holds: $$\lim_{x \rightarrow \infty} \dfrac{\pi(x)}{\mathrm{li}(x)}=1,$$ where $\pi$ denotes the prime counting function and $\mathrm{li}$ denotes the logarithmic integral.
