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, defineThen for sufficiently large n, the nth prime number, pn, is bounded above and below byThe 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.

This width, relative to pn, decreases something like 1/log³ n.


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