Logarithmic integral function

Definite integral defined as:

∫₀^x 1/ln t dt

is called logarithmic integral or integral logarithm and denoted with li(x) or Li(x). For x > 1 in a point t=1 this integral diverges, in this case we use for Li(x) the main value of unessential integral. Logarithmic integral comes in a variety of formulas concerning the density of primes in number theory and specially in Prime numbers theorem, where for example the estimation for prime counting function π(n) is:

π(n) ~ Li(n) = ∫₂ⁿ 1/ ln t dt.

This integral is in a connection with integral exponential function such as that li(x) = Ei (ln x).