# Bounds on the nth prime

The nth prime is approximately n log n.

For more precise estimates, there are numerous upper and lower bounds for the nth prime, each tighter over some intervals than others. Here I want to point out upper and lower bounds from a dissertation by Christian Axler on page viii.

First, define

Then for sufficiently large n, the nth prime number, pn, is bounded above and below by

The lower bound holds for n ≥ 2, and the upper bound holds for n ≥ 8,009,824.

The width of the bracket bounding pn is 0.787 n / log²n.

The bracket grows roughly linearly with n, but the primes grow like n log n, and so the width of the bracket relative to pn decreases like 1/log n.