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