Definite integral defined as:
- ∫0x 1/ln t dt
is a non-elemental function 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 with the main value of nondefinite 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). If we substitute x with eu, we get a serie:
- li(eu) = γ + ln u + u + u2/2 · 2! + u3/3 · 3! + u4/4 · 4! - ...,
where γ ≈ 0.57721 56649 01532 is Euler-Mascheroni's constant.