  We study the distribution of lattice points with prime coordinates lying in
the dilate of a convex planar domain having smooth boundary, with nowhere
vanishing curvature. Counting lattice points weighted by a von Mangoldt
function gives an asymptotic formula, with the main term being the area of the
dilated domain, and our goal is to study the remainder term. Assuming the
Riemann Hypothesis, we give a sharp upper bound, and further assuming that the
positive imaginary parts of the zeros of the Riemann zeta functions are
linearly independent over the rationals allows us to give a formula for the
value distribution function of the properly normalized remainder term.
