  The active galactic nuclei X-ray luminosity function traces actively
accreting supermassive black holes and is essential for the study of the
properties of the active galactic nuclei (AGN) population, black hole
evolution, and galaxy-black hole coevolution. Up to now, the AGN luminosity
function has been estimated several times in soft (0.5-2 keV) and hard X-rays
(2-10 keV). AGN selection in these energy ranges often suffers from
identification and redshift incompleteness and, at the same time, photoelectric
absorption can obscure a significant amount of the X-ray radiation. We estimate
the evolution of the luminosity function in the 5-10 keV band, where we
effectively avoid the absorbed part of the spectrum, rendering absorption
corrections unnecessary up to NH=10^23 cm^-2. Our dataset is a compilation of
six wide, and deep fields: MAXI, HBSS, XMM-COSMOS, Lockman Hole, XMM-CDFS,
AEGIS-XD, Chandra-COSMOS, and Chandra-CDFS. This extensive sample of ~1110 AGN
(0.01<z<4.0, 41<log L_x<46) is 98% redshift complete with 68% spectroscopic
redshifts. We use Bayesian analysis to select the best parametric model from
simple pure luminosity and pure density evolution to more complicated
luminosity and density evolution and luminosity-dependent density evolution. We
estimate the model parameters that describe best our dataset separately for
each survey and for the combined sample. We show that, according to Bayesian
model selection, the preferred model for our dataset is the
luminosity-dependent density evolution (LDDE). Our estimation of the AGN
luminosity function does not require any assumption on the AGN absorption and
is in good agreement with previous works in the 2-10 keV energy band based on
X-ray hardness ratios to model the absorption in AGN up to redshift three. Our
sample does not show evidence of a rapid decline of the AGN luminosity function
up to redshift four. [abridged]
