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

TL;DR

This paper establishes a reciprocity formula for a mixed second moment involving the Riemann zeta function and Dirichlet L-functions, connecting automorphic forms and analytic number theory.

Contribution

It proves Motohashi's formula for mixed moments involving zeta and Dirichlet L-functions, extending previous results to new cases and methods.

Findings

Derived a new reciprocity formula for mixed moments

Connected automorphic forms with moments of L-functions

Confirmed consistency with earlier Motohashi results for prime moduli

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.