Logarithmic integral function

In mathematics, the logarithmic integral function or integral logarithm li(x) is a non-elementary function defined for all positive real numbers x≠ 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:

Sometimes instead of li the offset logarithmic integral is used, defined as
. This is often used in number theoretic applications. Neither function should be confused with
the logarithmic integral whose definition is
:.

The growth behavior of this function for x → ∞ is

(see big O notation).

The logarithmic integral finds application in many areas, in particular it is used is in estimates of prime number densities, such as the prime number theorem:

π(x) ~ li(x) ~ Li(x)

where π(x) denotes the number of primes smaller than or equal to x.

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

li(x) = Ei (ln (x)) for all positive real x ≠ 1.

This leads to series expansions of li(x), for instance:

where γ ≈ 0.57721 56649 01532 ... is the Euler-Mascheroni gamma constant. 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.

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