We present the first direct measurements of the rest-frame 10-40 keV X-ray luminosity function (XLF) of Active Galactic Nuclei (AGNs) based on a sample of 94 sources at 0.1 < z <3, selected at 8-24 keV energies from sources in the NuSTAR extragalactic survey program. Our results are consistent with the strong evolution of the AGN population seen in prior, lower-energy studies of the XLF. However, different models of the intrinsic distribution of absorption, which are used to correct for selection biases, give significantly different predictions for the total number of sources in our sample, leading to small, systematic differences in our binned estimates of the XLF. Adopting a model with a lower intrinsic fraction of Compton-thick sources and a larger population of sources with column densities N_H ~ 10^{23-24} /cm2 or a model with stronger Compton reflection component (with a relative normalization of R ~ 2 at all luminosities) can bring extrapolations of the XLF from 2-10 keV into agreement with our NuSTAR sample. Ultimately, X-ray spectral analysis of the NuSTAR sources is required to break this degeneracy between the distribution of absorbing column densities and the strength of the Compton reflection component and thus refine our measurements of the XLF. Furthermore, the models that successfully describe the high-redshift population seen by NuSTAR tend to over-predict previous, high-energy measurements of the local XLF, indicating that there is evolution of the AGN population that is not fully captured by the current models.
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