Logarithmic integral

In some 'esoteric' areas of mathematics, 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:

<math> {\rm li} (x) = \int_{0}^{x} \frac{dt}{\ln (t)} \; . </math>

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:

<math> {\rm li} (x) = \lim_{\varepsilon \to 0} \left( \int_{0}^{1-\varepsilon} \frac{dt}{\ln (t)} + \int_{1+\varepsilon}^{x} \frac{dt}{\ln (t)} \right) \; . </math>

The growth behavior of this function for x → ∞ is

<math> {\rm li} (x) = \Theta \left( {x\over \ln (x)} \right) \; . </math>

(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 a multiplicative function - the number of primes smaller than or equal to x, and Li(x) is the offset logarithmic integral function, related to li(x) by Li(x) = li(x) - li(2).

The offset logarithmic integral gives a slightly better estimate to the π function than li(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:

<math> {\rm li} (e^{u}) = \gamma + \ln \left| (u) \right| + \sum_{n=1}^{\infty} {u^{n}\over n \cdot n!} \quad {\rm for} \; u \ne 0 \; , </math>

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.