  A new derivation of the classic asymptotic expansion of the n-th prime is
presented. A fast algorithm for the computation 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 retained the
first m terms, for $1\le m\le 11$, are given. Finally, assuming the Riemann
Hypothesis, we give estimations of the best possible $r_3$ such that, for $n\ge
r_3$, we have $p_n> s_3(n)$ where $s_3(n)$ is the sum of the first four terms
of the asymptotic expansion.
