Offset logarithmic integral

In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number theoretic significance, occurring in the prime number theorem as an estimate of the number of prime numbers less than a given value.

The logarithmic integral has an integral representation 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 a Cauchy principal value,

The offset logarithmic integral or Eulerian logarithmic integral is defined as

or, integrally represented

As such, the integral representation has the advantage of avoiding the singularity in the domain of integration.

This function is a very good approximation to the number of prime numbers less than x.

The function li(x) is related to the exponential integral Ei(x) via the equation

which is valid for x > 0. This identity provides a series representation of li(x) as

where γ ≈ 0.57721 56649 01532 ... is the Euler–Mascheroni gamma constant. A more rapidly convergent series due to Ramanujan is