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

TL;DR

The paper establishes a Motohashi-type reciprocity for the mixed second moment $\mathcal{M}_2(s_1,s_2,s_3,s_4; g; \psi)$ of $\zeta$ and a Dirichlet $L$-function. It shows that, after meromorphic continuation, $\mathcal{M}_2$ equals an explicitly computable main term $\mathcal{N}$ plus a dual cubic moment, which factors into Maaß, Eisenstein, and holomorphic contributions and involves central values $L(\tfrac{1}{2}, f)^2 L(\tfrac{1}{2}, f\otimes \overline{\psi})$. The proof blends approximate functional equations, Voronoï summation, and the Kuznetsov formula to transfer off-diagonal sums to a spectral cubic moment on $\mathrm{GL}_2$, and analyzes all local factors to obtain explicit kernel expressions. Consequently, the results recover Motohashi’s original reciprocity for odd prime $q$ and extend it to all $q$, with corollaries giving short-interval bounds for the mixed moment, yielding near-optimal estimates in terms of $H$, $q$, and $T$. The work highlights a deep link between second moments of zeta-type objects and cubic moments of automorphic $L$-functions, with implications for subconvex-type bounds in short intervals.

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 $Γ_{0}(q) \backslash \mathbb{H}$. The methods involve a blend of analytic number theory and automorphic forms.

Theorems & Definitions (20)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Theorem 3.1
• proof
• Lemma 3.2: Twisted Ramanujan expansion
• proof
• Lemma 3.3
• proof
• Lemma 3.4