Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$Functions and Applications
Abstract
A new reciprocity formula for Dirichlet $L$functions associated to an arbitrary primitive Dirichlet character of prime modulus $q$ is established. We find an identity relating the fourth moment of individual Dirichlet $L$functions in the $t$aspect to the cubic moment of central $L$values of HeckeMaass newforms of level at most $q^{2}$ and primitive central character $\psi^{2}$ averaged over all primitive nonquadratic characters $\psi$ modulo $q$. Our formulae would be viewed as reverse versions of recent work of PetrowYoung. Direct corollaries include a version of Iwaniec's short interval fourth moment bound and the twelfth moment bound for individual Dirichlet $L$functions, which generalise work of Jutila and JutilaMotohashi, respectively. This work traverses an intersection of analytic number theory and automorphic forms.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.08974
 Bibcode:
 2021arXiv211008974K
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Representation Theory;
 11M06;
 11F72 (primary);
 11F03 (secondary)
 EPrint:
 38 pages