Daniel Fiorilli (Université Paris-Sud)
Since the work of Selberg and of Barban, Davenport and Halberstam, the variances of primes in intervals and in arithmetic progressions have been widely studied and continue to be an active topic of research. However, much less is known about higher moments. Hooley established a bound on the third moment in progressions, which was later sharpened by Vaughan for a variant involving a major arcs approximation.
Little is known for moments of order four or higher, other than the conjecture of Hooley and the conditional result of Montgomery-Soundararajan. In this talk I will discuss recent joint work with Régis de la Bretèche on weighted moments in intervals and on weighted moments of moments in progressions. In particular we will show how to deduce sharp unconditional omega results on all weighted even moments in certain ranges.