  We establish unconditional $\Omega$-results for all weighted even moments of
primes in arithmetic progressions. We also study the moments of these moments
and establish lower bounds under GRH. Finally, under GRH and LI we prove an
asymptotic for all moments of the associated limiting distribution, which in
turn indicates that our unconditional and GRH results are essentially best
possible. Using our probabilistic results, we formulate a conjecture on the
moments with a precise associated range of validity, which we believe is also
best possible. This last conjecture implies a $q$-analogue of the
Montgomery-Soundararajan conjecture on the Gaussian distribution of primes in
short intervals. The ideas in our proofs include a novel application of
positivity in the explicit formula and the combinatorics of arrays of
characters which are fixed by certain involutions.

Number Theory