The $n$-th prime asymptotically

A new derivation of the classic asymptotic expansion
of the n-th prime is presented. A fast algorithm for the compu-
tation of its terms is also given, which will be an improvement of
that by Salvy (1994).
Realistic bounds for the error with $li−1 (n)$, after having re-
tained the first $m$ terms, for $1 ≤ m ≤ 11$, are given. Finally, as-
suming the Riemann Hypothesis, we give estimations of the best
possible $r_3$ such that, for $n ≥ r_3$ , we have $p_n > s_3 (n)$ where $s_3 (n)$
is the sum of the first four terms of the asymptotic expansion.