Continuous statistics of the Lyman-alpha forest at 0 < z < 1.6: the mean flux, flux distribution, and autocorrelation from HST FOS spectra

We measure the amount of absorption in the Lyman-alpha forest at 0 < z < 1.6 in HST FOS spectra of 74 QSOs. At 0 < z < 1.6 we find that 79% of the absorption is from the low density intergalactic medium, 12% from metals and 9% from the strong H I lines, nearly identical to the percentages (78, 15 and 7) that we measured independently at z=2 from spectra taken with the Kast spectrograph on the Lick 3-m. At z=1 the low density intergalactic medium absorbs 0.037 +/- 0.004 of the flux. The error includes some but not all of the uncertainty in the continuum level. The remaining part gives relative errors of approximately 0.21 when we report the mean absorption in eight independent redshift intervals, and 0.047 when we average over all redshifts. We find 1.46 times more absorption from the low density intergalactic medium than comes from Ly-alpha lines that Bechtold et al. 2002 listed in the same spectra. The amount of absorption increases with z and can be fit by a power law in (1+z) with index 1.01. This corresponds to no change in the number of lines, of fixed rest frame equivalent widths, per unit redshift, consistent with the Janknecht et al. 2006 results on the distribution of lines. When we include similar measurements from higher redshifts, we need more degrees of freedom to fit the amount of absorption at 0 < z < 3.2. A power law with a break in slope, changing from index 1.5 at low z to 3.0 above z ~ 1.1 is a better but only marginally acceptable fit. We also calculate two other continuous statistics, the flux probability distribution function and the flux autocorrelation function that is non zero out to v ~ 500 km/sec at 0.5 < z < 1.5