Moments of moments of primes in arithmetic progressions

Abstract

AbstractWe establish unconditional ‐results for all weighted even moments of primes in arithmetic progressions. We also study the moments of these moments and establish lower bounds under the Generalized Riemann Hypothesis (GRH). Finally, under GRH and the Linear Independence Hypothesis (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 ‐analog 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 that are fixed by certain involutions.