Logarithmic integral function

The logarithmic integral or integral logarithm li(x) is a non-elementary function defined for all positive real numbers x≠ 1 by the definite integral:

li(x) = ₀∫^x 1/ln t dt.

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 Cauchy's principal value:

li(x) = lim_{ε→0 0}∫^1-ε 1/ln t dt + _1+ε∫^x 1/ln t dt.

The logarithmic integral is mainly important because it occurs in estimates of prime number densities, especially in the prime number theorem.

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:

li(e^u) = γ + ln |u| + _n=1 ∑^∞ uⁿ/(n · n!) for u ≠ 0

where γ ≈ 0.57721 56649 01532 ... is Euler-Mascheroni's constant and

li(x^m) = γ + ln |ln x| - ln m + _n=1 ∑^∞ (ln x)ⁿ mⁿ/(n n!).

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.