The *n*th prime is approximately *n* log *n*.

For more precise estimates, there are numerous upper and lower bounds for the *n*th 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 *n*th prime number, *p*_{n}, 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 *p*_{n} is 0.787 *n* / log²*n*.

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