Cody Problem 241, which is based on Project Euler Problem 7, asks us to identify the Nth prime number. That is, the problem seeks the inverse of the prime counting function pi(n), which provides the number of primes less than or equal to n. The Prime Number Theorem gives approximate forms of pi(n) for large n. Two such approximations are n/ln(n) and the offset logarithmic integral Li(n) = li(n) - li(2), where li(x) = integral of 1/ln(t) from t = 0 to t = infinity (See Cody Problem 46066).
Test these approximations by computing two ratios: r1 = [n/ln(n)]/pi(n) and r2 = Li(n)/pi(n). Do not round the approximations to integers. For n = 100, you will find that the first approximation is about 13% low and the second is about 16% high. However, for n = 10^8, the first approximation is 6% low and the second is only 0.01% high.
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Chris Cleveland George Berken Augusto Mazzei Christian Schröder

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