logarithmic integral (idea)

The logarithmic integral, or Li(x), is the integral from zero to x of 1/ln(x), where ln(x) is the natural logarithm of x. This is commonly displayed like so:

         / x
        /
       |
        \       dt
Li(x)=   \    ------
          |    ln(t)
         /
        /  0

The prime number theorem states that:

       ln(x)pi(x)
 lim  ------------- = 1
x->∞        x

or, equivalently,

        pi(x)
 lim   ------- = 1
x->∞    Li(x)

where pi(x) is the number of primes less than or equal to x
(with the exception that for the prime number theorem,
Li(x) is the integral from 2 to x instead of from 0 to x).

Li(x) is an elliptic integral because there is no "elementary" function which it is equivalent to. In other words, it can at best be approximated.

Because of its relation to the prime number theorem, the logarithmic integral is also related to the Riemann zeta function.

Skewes number is an upper bound on the first time for which Li(x)<pi(x); it has been proven that Li(x) switches between being greater than and less than pi(x) an infinite number of times.