  Finding and characterizing the population of active galactic nuclei (AGNs)
that produces the X-ray background (XRB) is necessary to connect the history of
accretion to observations of galaxy evolution at longer wavelengths. The year
2012 will see the deployment of the first hard X-ray imaging telescope that,
through deep extragalactic surveys, will be able to measure the AGN population
at the energies where the XRB peaks (~20-30 keV). Here, we present predictions
of AGN number counts in three hard X-ray bandpasses: 6-10 keV, 10-30 keV and
30-60 keV. Separate predictions are presented for the number counts of Compton
thick AGNs, the most heavily obscured active galaxies. The number counts are
calculated for five different models of the XRB that differ in the assumed hard
X-ray luminosity function, the evolution of the Compton thick AGNs, and the
underlying AGN spectral model. The majority of the hard X-ray number counts
will be Compton thin AGNs, but there is a >10x increase in the Compton thick
number counts from the 6-10 keV to the 10-30 keV band. The Compton thick
population show enough variation that a hard X-ray number counts measurement
will constrain the models. The computed number counts are used to consider
various survey strategies for the NuSTAR mission, assuming a total exposure
time of 6.2 Ms. We find that multiple surveys will allow a measurement of
Compton thick evolution. The predictions presented here should be useful for
all future imaging hard X-ray missions.
