The Logarithmic Integral
The logarithmic integral, Li(x), is defined as:
where the integral is to be understood as a Cauchy Principal Value integral.
This definition is extended to complex arguments z via the formula Li⁡x=Ei⁡ln⁡x. Note that the resulting branch cuts are the intervals −∞,0 and ⁡0,1. However, since Li⁡x is defined as a Cauchy principal value integral, the values on the branch cuts are "isolated". That is, the complex function Li⁡z is not continuous onto the branch cuts from either above or below.
Li(x) provides an approximation to the number of primes less than or equal to x.
and the actual number of primes≤1000 is:
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