  We develop a method to calculate the column density distribution of the
Lyman-alpha forest for column densities in the range $10^{12.5} - 10^{14.5}
cm^{-2}$. The Zel'dovich approximation, with appropriate smoothing, is used to
compute the density and peculiar velocity fields. The effect of the latter on
absorption profiles is discussed and it is shown to have little effect on the
column density distribution. An approximation is introduced in which the column
density distribution is related to a statistic of density peaks (involving its
height and first and second derivatives along the line of sight) in real space.
We show that the slope of the column density distribution is determined by the
temperature-density relation as well as the power spectrum on scales $2 h
Mpc^{-1} < k < 20 h Mpc^{-1}$. An expression relating the three is given. We
find very good agreement between the column density distribution obtained by
applying the Voigt-profile-fitting technique to the output of a full
hydrodynamic simulation and that obtained using our approximate method for a
test model. This formalism then is applied to study a group of CDM as well as
CHDM models. We show that the amplitude of the column density distribution
depends on the combination of parameters $(\Omega_b h^2)^2 T_0^{-0.7}
J_{HI}^{-1}$, which is not well-constrained by independent observations. The
slope of the distribution, on the other hand, can be used to distinguish
between different models: those with a smaller amplitude and a steeper slope of
the power spectrum on small scales give rise to steeper distributions, for the
range of column densities we study. Comparison with high resolution Keck data
is made.
