Logarithmic integral

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The logarithmic integral is $$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{1}{\log(t)} \mathrm{d}t,$$ 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

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