  Radio source counts constrain galaxy populations and evolution, as well as
the global star formation history. However, there is considerable disagreement
among the published 1.4-GHz source counts below 100 microJy. Here we present a
statistical method for estimating the microJy and even sub-microJy source count
using new deep wide-band 3-GHz data in the Lockman Hole from the Karl G. Jansky
Very Large Array (VLA). We analyzed the confusion amplitude distribution P(D),
which provides a fresh approach in the form of a more robust model, with a
comprehensive error analysis. We tested this method on a large-scale
simulation, incorporating clustering and finite source sizes. We discuss in
detail our statistical methods for fitting using Monte Carlo Markov chains,
handling correlations, and systematic errors from the use of wide-band radio
interferometric data. We demonstrated that the source count can be constrained
down to 50 nJy, a factor of 20 below the rms confusion. We found the
differential source count near 10 microJy to have a slope of -1.7, decreasing
to about -1.4 at fainter flux densities. At 3GHz the rms confusion in an
8arcsec FWHM beam is ~ 1.2 microJy/beam, and a radio background temperature ~
14mK. Our counts are broadly consistent with published evolutionary models.
With these results we were also able to constrain the peak of the Euclidean
normalized differential source count of any possible new radio populations that
would contribute to the cosmic radio background down to 50 nJy.
