  We develop an analytical method based on the lognormal approximation to
compute the column density distribution of the Lyman-alpha forest in the low
column density limit. We compute the column density distributions for six
different cosmological models and found that the standard, COBE-normalized CDM
model cannot fit the observations of the Lyman-alpha forest at z=3. The
amplitude of the fluctuations in that model has to be lowered by a factor of
almost 3 to match observations. However, the currently viable cosmological
models like the lightly tilted COBE-normalized CDM+Lambda model, the CHDM model
with 20% neutrinos, and the low-amplitude Standard CDM model are all in
agreement with observations, to within the accuracy of our approximation, for
the value of the cosmological baryon density at or higher than the old Standard
Bing Bang Nucleosynthesis value of 0.0125 for the currently favored value of
the ionizing radiation intensity. With the low value for the baryon density
inferred by Hogan & Rugers (1996), the models can only marginally match
observations.
