On the error term in a mixed moment of L-functions

TL;DR

This work improves the error term in a mixed moment of automorphic L-functions twisted by Dirichlet characters, achieving $\displaystyle \frac{1}{\varphi^*(q)}\sum^*_{\chi\bmod q} L(\tfrac{1}{2}, f\otimes \chi)\overline{L(\tfrac{1}{2}, \chi)}^2 = \frac{L(1,f)^2}{\zeta(2)} + O(q^{-\eta_f+\epsilon})$ with explicit $\eta_f$ values $\eta_f=\tfrac{1}{22}$ for holomorphic $f$ and $\eta_f=\tfrac{5}{152}$ for Maass $f$ (and a related symmetric case with $\delta>0$). The authors introduce a new bound for a structured sum $C^{\pm}(M,N,N_1,N_2)$ that leverages additive-detection, Poisson summation, and Wilton-type cancellation to achieve square-root saving in the critical range $N/M\sim q^{1/2}$, and combine this with a switching trick and Voronoi summation to reduce the problem to a small family of parameter ranges. The resulting bound is incorporated into the analysis for the cases $M>N$ and $N\ge M$, yielding a substantially improved error term that matches the quality seen in the fourth moment for Dirichlet L-functions and sharpens previous results for the mixed moment. This has potential implications for related moments and the understanding of oscillatory sums arising in L-function analytics.

Abstract

There has recently been some interest in optimizing the error term in the asymptotic for the fourth moment of Dirichlet L-functions and a closely related mixed moment of L-functions involving automorphic L-functions twisted by Dirichlet characters. We obtain an improvement for the error term of the latter.

Theorems & Definitions (8)

• Theorem 1.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Lemma 3.7