  We present a new, uniform analysis of the HI transmitted flux (F) and HI
column density (N(HI)) distribution in the low-density IGM as a function of
redshift z for 0<z<3.6 using 55 HST/COS FUV (Delta(z)=7.2 at z<0.5), five
HST/STIS+COS NUV (Delta(z)=1.3 at z~1) and 24 VLT/UVES and Keck/HIRES
(Delta(z)=11.6 at 1.7<z<3.6) AGN spectra. We performed a consistent, uniform
Voigt profile analysis to combine spectra taken with different instruments, to
reduce systematics and to remove metal-line contamination. We confirm
previously known conclusions on firmer quantitative grounds in particular by
improving the measurements at z~1. Two flux statistics at 0<F<1, the mean HI
flux and the flux probability distribution function (PDF), show that
considerable evolution occurs from z=3.6 to z=1.5, after which it slows down to
become effectively stable for z<0.5. However, there are large sightline
variations. For the HI column density distribution function (CDDF, f proptional
to N(HI)^(-beta)) at log (N(HI)/1cm^-2)=[13.5, 16.0], beta increases as z
decreases from beta~1.60 at z~3.4 to beta~1.82 at z~0.1. The CDDF shape at
lower redshifts can be reproduced by a small amount of clockwise rotation of a
higher-z CDDF with a slightly larger CDDF normalisation. The absorption line
number per z (dn/dz) shows a similar evolutionary break at z~1.5 as seen in the
flux statistics. High-N(HI) absorbers evolve more rapidly than low-N(HI)
absorbers to decrease in number or cross-section with time. The individual
dn/dz shows a large scatter at a given z. The scatter increases toward lower z,
possibly caused by a stronger clustering at lower z.
