Catinca Mujdei

An asymptotic formula for the (non-holomorphic) Conrey--Iwaniec cubic moment averaged over a neighbourhood

Conrey and Iwaniec established a Lindelöf-on-average estimate for the cubic moment of central values of L-functions attached to cusp forms of even weight at least 12, twisted by a quadratic character. The result was later extended to arbitrary even weight by Petrow, who obtained in particular an expression for the cubic moment that is reminiscent of some formulas obtained by Motohashi. Petrow's formula consists of a main term plus some dual moment, providing encouraging evidence towards the asymptotic formula conjectured by Conrey--Farmer--Keating--Rubinstein--Snaith for the cubic moment. The best bound obtained by current techniques for the dual moment is unfortunately a log-power larger than the main term, so the Motohashi-type formula provides only an ''almost asymptotic'' formula for the cubic moment. The subject of this talk will be the non-holomorphic analogue of this cubic moment, which exhibits the same behaviour. I will explain how averaging over an appropriate neighbourhood of characters shortens the dual moment, thus getting a small step closer to the conjectured asymptotic formula.