 In particular, according to the Siegel-Walfisz theorem it is a very good approximation to the prime-counting function, which is defined as the number of prime numbers less than or equal to a given value formula_1.The logarithmic integral has an integral representation defined for all positive real numbers  ≠ 1 by the definite integral

Here, denotes the natural logarithm.The function has a singularity at  = 1, and the integral for  > 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.