We present the highest quality Lyman Alpha forest spectra published to date, from the QSO HS 1946+7658. The distribution of H I column densities is a power law of slope -1.5 from Log N = 12.1 - 14. This power law can extend to N = 0, because lines weaker than Log N = 12.1 do not have a large H I optical depth. Low column lines with Log N > 9 could account for all observed He II absorption, but lines with Log N > 12 alone are unlikely to do so. The b distribution between 20 and 60 km/sec is a Gaussian with a mean of 23 km/sec (less than reported in past at this z), and a sigma b of 14 km/sec. We report no evolution in the Lyman alpha forest (except the number of lines), because Lu et al. (1997) found the same N and b distributions at <z> = 3.7. We see lines with 14 < b < 20 km/sec and b > 80 km/sec that cannot be accounted for by noise or blending effects. We discover that the lower cutoff in the b distribution varies with N, from b = 14 km/sec at Log N = 12.5 to b = 22 km/sec at Log N = 14.0, but otherwise b and N are not correlated. We see no Lyman Alpha line clustering above 50 kms, in disagreement with previous results from lower signal to noise data, but we do see a 3 sigma clustering signal at 25 - 50 km/sec among lines with Log N > 13.6
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