Motohashi's formula for the fourth moment of individual Dirichlet L-functions and applications

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 Hecke-Maass newforms of level at most q(2) and primitive central character psi(2) averaged over all primitive nonquadratic characters psi modulo q. 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 Dirichlet L-functions, which generalise work of Jutila and Heath-Brown, respectively. This work traverses an intersection of classical analytic number theory and automorphic forms.