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The Serendipitous Extragalactic X-ray Source Identification (SEXSI) Programis designed to extend greatly the sample of identified extragalactic hard X-ray2-10 keV sources at intermediate fluxes ($\sim 10^{-13} - 10^{-15} erg/cm2/s$).SEXSI, which studies sources selected from more than 2 deg$^2$, provides anessential complement to the {\em Chandra} Deep Fields, which reach depths of $5\times 10^{-16} erg/cm2/s$ (\hardrange) but over a total area of $< 0.2$deg$^2$. In this paper we describe the characteristics of the survey and ourX-ray data analysis methodology. We present the cumulative flux distributionfor the X-ray sample of 1034 hard sources, and discuss the distribution ofspectral hardness ratios. Our lognlogs in this intermediate flux range connectsto those found in the deep fields, and by combining the data sets, we constrainthe hard X-ray population over the flux range where the differential numbercounts change slope, and from which the bulk of the 2 -- 10 keV X-raybackground arises. We further investigate the lognlogs distribution separatelyfor soft and hard sources in our sample, finding that while a clear change inslope is seen for the softer sample, the hardest sources are well-described bya single power-law down to the faintest fluxes, consistent with the notion thatthey lie at lower average redshift.Comment: 33 pages, accepted to Ap

The Serendipitous Extragalactic X‐Ray Source Identification Program. I. Characteristics of the Hard X‐Ray Sample