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We present the results of a survey for super Lyman limit systems (SLLS;defined to be absorbers with 19.0 <= log(NHI) <= 20.3 cm^-2) from a largesample of high resolution spectra acquired using the Keck and Magellantelescopes. Specifically, we present 47 new SLLS from 113 QSO sightlines. Wefocus on the neutral hydrogen frequency distribution f(N,X) of the SLLS and itsmoments, and compare these results with the Lyman-alpha forest and the dampedLyman alpha systems (DLA; absorbers with log(NHI) >= 20.3 cm^-2). We find thatthat f(N,X) of the SLLS can be reasonably described with a power-law of indexalpha = -1.43^{+0.15}_{-0.16} or alpha = -1.19^{+0.20}_{-0.21} depending onwhether we set the lower N(HI) bound for the analysis at 10^{19.0} cm^-2 or10^{19.3}$ cm^-2, respectively. The results indicate a flattening in the slopeof f(N,X) between the SLLS and DLA. We find little evidence for redshiftevolution in the shape of f(N,X) for the SLLS over the redshift range of thesample 1.68 < z < 4.47 and only tentative evidence for evolution in the zerothmoment of f(N,X), the line density l_lls(X). We introduce the observabledistribution function O(N,X) and its moment, which elucidates comparisons of HIabsorbers from the Lyman-alpha through to the DLA. We find that a simple threeparameter function can fit O(N,X) over the range 17.0 <= log(NHI) <=22.0. Weuse these results to predict that f(N,X) must show two additional inflectionsbelow the SLLS regime to match the observed f(N,X) distribution of theLyman-alpha forest. Finally, we demonstrate that SLLS contribute a minorfraction (~15%) of the universe's hydrogen atoms and, therefore, an even smallfraction of the mass in predominantly neutral gas.Comment: 15 pages, 10 figures, accepted to the Astrophysical Journal. Revision  includes updated reference

The Keck+Magellan Survey for Lyman Limit Absorption. I. The Frequency Distribution of Super Lyman Limit Systems