Mixed Moments of the Riemann Zeta and Dirichlet $L$-Functions

Abstract

We prove Motohashi's formula for a mixed second moment of the Riemann zeta function and a Dirichlet $L$-function attached to a primitive Dirichlet character modulo $q\in \mathbb{N}$. If $q$ is an odd prime, our reciprocity formula is consistent with Motohashi's result in the early 1990s. The cubic moment side features two versions of central $L$-values of automorphic forms on ${\Gamma }_0(q)\backslash \mathbb{H}$. The methods involve a blend of analytic number theory and automorphic forms.

20 pages. LaTeX2e