Logarithmic integral function

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In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It occurs in problems of physics and has number theoretic significance, occurring in the prime number theorem as an estimate of the number of prime numbers less than a given value.

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Integral representation

The logarithmic integral has an integral representation defined for all positive real numbers x\ne 1 by the definite integral:

Here, ln denotes the natural logarithm. The function 1 / ln(t) has a singularity at t = 1, and the integral for x > 1 has to be interpreted as a Cauchy principal value:

Offset logarithmic integral

The offset logarithmic integral or Eulerian logarithmic integral is defined as

or

As such, the integral representation has the advantage of avoiding the singularity in the domain of integration.

This function is a very good approximation to the number of prime numbers less than x.

Series representation

The function li(x) is related to the exponential integral Ei(x) via the equation

which is valid for x > 1. This identity provides a series representation of li(x) as

where γ ≈ 0.57721 56649 01532 ... is the Euler–Mascheroni gamma constant. A more rapidly convergent series due to Ramanujan [1] is

Special values

The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 ...; this number is known as the Ramanujan–Soldner constant.

li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151…

This is -(\Gamma\left(0,-\ln 2\right) + i\,\pi) where \Gamma\left(a,x\right) is the incomplete gamma function. It must be understood as the Cauchy principal value of the function.

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