Prime number theorem
Approximations for the n th prime number

As a consequence of the prime number theorem, one gets an asymptotic expression for the n th prime number, denoted by p:

A better approximation is

Again considering the 2 × 10 th prime number 8 512 677 386 048 191 063, this gives an estimate of 8 512 681 315 554 715 386; the first 5 digits match and relative error is about 0.00005%.

Rosser's theorem states that

This can be improved by the following pair of bounds:
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