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.
Referenced By
Elementary function | Elementary functions | List of functions | List of mathematical functions | List of mathematical topics (J-L) | Special function | Special functions
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