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.