Difference between revisions of "Logarithmic integral"

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File:Liplot.png|Graph of $\mathrm{li}$.
 
File:Liplot.png|Graph of $\mathrm{li}$.
File:Domain coloring of log integral.png|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{li}$.
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File:Complexliplot.png|[[Domain coloring]] of $\mathrm{li}$.
 
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Revision as of 21:18, 23 May 2016

The logarithmic integral is $$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{dt}{\log(t)},$$ where $\log$ denotes the logarithm.

Properties

Theorem

The following formula holds: $$\mathrm{li}(x)=\mathrm{Ei}( \log(x)),$$ where $\mathrm{li}$ denotes the logarithmic integral, $\mathrm{Ei}$ denotes the exponential integral Ei, and $\log$ denotes the logarithm.

Proof

References

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.

Proof

References

See Also

Prime counting function