Motohashi’s formula for the fourth moment of individual DirichletL-functions and applications

Abstract

AbstractA new reciprocity formula for DirichletL-functions associated to an arbitrary primitive Dirichlet character of prime modulusqis established. We find an identity relating the fourth moment of individual DirichletL-functions in thet-aspect to the cubic moment of centralL-values of Hecke–Maaß newforms of level at most$q^{2}$and primitive central character$\psi ^{2}$averaged over all primitive nonquadratic characters$\psi $moduloq. Our formula can be thought of as a reverse version of recent work of Petrow–Young. Direct corollaries involve a variant of Iwaniec’s short interval fourth moment bound and the twelfth moment bound for DirichletL-functions, which generalise work of Jutila and Heath-Brown, respectively. This work traverses an intersection of classical analytic number theory and automorphic forms.