We present a study of the evolution of the column density distribution, f(N,z), and total neutral hydrogen mass in high-column density quasar absorbers using candidates from a recent high-redshift survey for damped Lyman-alpha (DLA) and Lyman limit system (LLS) absorbers. The observed number of LLS (N(HI)> 1.6 * 10^{17} atom/cm^2) is used to constrain f(N,z) below the classical DLA Wolfe et al. (1986) definition of 2 * 10^{20} atom/cm^2. The joint LLS-DLA analysis shows unambiguously that f(N,z) deviates significantly from a single power law and that a Gamma-law distribution of the form f(N,z)=(f_*/N_*)(N/N_*)^{-Beta} exp(-N/N_*) provides a better description of the observations. These results are used to determine the amount of neutral gas contained in DLAs and in systems with lower column density. Whilst in the redshift range 2 to 3.5, ~90% of the neutral HI mass is in DLAs, we find that at z>3.5 this fraction drops to only 55% and that the remaining missing mass fraction of the neutral gas lies in sub-DLAs with N(HI) 10^{19} - 2 * 10^{20} atom/cm^2. The characteristic column density, N_*, changes from 1.6 * 10^{21} atom/cm^2 at z<3.5 to 2.9 * 10^{20} atom/cm^2 at z>3.5, supporting a picture where at z>3.5, we are directly observing the formation of high column density neutral hydrogen DLA systems from lower column density units. Moreover since current metallicity studies of DLA systems focus on the higher column density systems they may be giving a biased or incomplete view of global galactic chemical evolution at z>3. After correcting the observed mass in HI for the ``missing neutral gas the comoving mass density now shows no evidence for a decrease above z=2. (abridged)
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