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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:

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:

The growth behavior of this function for x → ∞ is

(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:

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.