Moments of moments of primes in arithmetic progressions

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