Logarithmic Integral Function

The logarithmic integral function, written li(x), is a special function defined for positive real numbers other than one by the integral from zero to x of dt over the natural log of t, interpreted through its Cauchy principal value when x is greater than one because the integrand has a singularity at t equals one. Its chief importance in mathematics is as an excellent approximation, under the prime number theorem, to the number of primes less than or equal to a given value x, which makes it central to the study of how prime numbers are distributed. The function can be extended into the complex plane, where it becomes multivalued with branch points at zero and one. Beyond number theory, where it connects to deep questions such as the Riemann hypothesis, the logarithmic integral also has computational uses in physics. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/