Offset logarithmic integral
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The offset logarithmic integral, or European logarithmic integral, is a non-elementary function Li(x) differing by a constant from the logarithmic integral function li(x), defined such that:
- <math> {\rm Li} (x) = \mathrm{li}(x) - \mathrm{li}(2).\,<math>
Explicitly, this means
- <math> {\rm Li} (x) = \int_{2}^{x} \frac{dt}{\ln t} \, <math>
where ln is the natural logarithm. It can be shown that
- <math> {\rm Li} (x) = \frac{x}{\ln x} \sum_{k=0}^{\infty} \frac{k!}{(\ln x)^k} <math>
or
- <math> \frac{{\rm Li} (x)}{x/\ln x} = 1 + \frac{1}{\ln x} + \frac{2}{(\ln x)^2} + \cdots. <math>
It is often used in formulations of the prime number theorem.