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

\begin{align*} f(n, k) &= \log n + \log \log n - 1 + \frac{\log \log n - 2}{\log n} \\ &\qquad - \frac{(\log\log n)^2 - 6\log\log n + k}{2\log^2 n} \end{align*}

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

n f(n, 11.847) <\, p_n < n f(n, 10.273)

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.