This paper uses the Rice method [18] to give bounds to
the distribution of the maximum of a smooth stationary Gaussian
process. We give simpler expressions of the first two terms of
the Rice series [3,13] for the distribution of the maximum.
Our main contribution is a simpler form of the second factorial moment
of the number of upcrossings which is in some sense a generalization
of Steinberg et al.'s formula
([7] p. 212).
Then, we present a numerical application and asymptotic expansions
that give a new interpretation of a result by
Piterbarg [15]
