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 growth behavior of this function for x → ∞ is

li(x) = Θ(x/ln(x))

(see big O notation).

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

π(x) ~ li(x)

where π(x) denotes the number of primes smaller than or equal to x. In order to avoid the principal value calculation, the prime number theorem is sometimes presented in terms of the integral ₂∫^x 1/ln t dt, which differs from li(x) by the value li(2) ≈ 1.04516.

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.

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.