# 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*.