  The classical heuristic complexity of the Number Field Sieve (NFS) is the
solution of an optimization problem that involves an unknown function, usually
noted $o(1)$ and called $\xi(N)$ throughout this paper, which tends to zero as
the entry $N$ grows. The aim of this paper is to find optimal asymptotic
choices of the parameters of NFS as $N$ grows, in order to minimize its
heuristic asymptotic computational cost. This amounts to minimizing a function
of the parameters of NFS bound together by a non-linear constraint. We provide
precise asymptotic estimates of the minimizers of this optimization problem,
which yield refined formulas for the asymptotic complexity of NFS. One of the
main outcomes of this analysis is that $\xi(N)$ has a very slow rate of
convergence: We prove that it is equivalent to
$4{\log}{\log}{\log}\,N/(3{\log}{\log}\,N)$. Moreover, $\xi(N)$ has an
unpredictable behavior for practical estimates of the complexity. Indeed, we
provide an asymptotic series expansion of $\xi$ and numerical experiments
indicate that this series starts converging only for $N>\exp(\exp(25))$, far
beyond the practical range of NFS. This raises doubts on the relevance of NFS
running time estimates that are based on setting $\xi=0$ in the asymptotic
formula.
