Really a 2 part question: What is the fastest way to find the nth prime and the inverse function $pi(n)$? And secondly what is the run-time of the algorithms?
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A question similar to your first question was already asked on this site. Most efficient algorithm for nth prime, deterministic and probabilistic?. And this answer seems to give information on your second question. – J.-E. Pin Dec 31 '16 at 17:47
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You could also use The nth Prime Page – J.-E. Pin Dec 31 '16 at 17:53