  We study a two-dimensional discrete directional maximal operator along the
set of the prime numbers. We show existence of a set of vectors, which are
lattice points in a sufficiently large annulus, for which the $\ell^2$ norm of
the associated maximal operator with supremum taken over all large scales grows
with an epsilon power in the number of vectors. This paper is a follow-up to a
prior work on the discrete directional maximal operator along the integers by
the first and third author.
