Secondary Term for the Mean Value of Maass Special $L$-values

TL;DR

This work proves a new secondary term in the mean value of Hecke–Maass L-values along an orthonormal basis of Maass cusp forms, namely $\sum_{t_f\le T} \omega_f L(\tfrac12+i t_f,f) = \frac{T^{2}}{\pi^{2}} + \frac{8 T^{3/2}}{3\pi^{3/2}} + O(T^{1+\varepsilon})$. The key method is an explicit formula for the smoothly weighted mean value, derived via a regularized Kuznetsov formula, Poisson summation, and Lerch zeta-transform techniques, enabling careful extraction of the main and secondary terms. The analysis relies on a detailed treatment of Bessel integrals through regularization, producing manageable diagonal and off-diagonal contributions and culminating in an explicit formula that yields the asymptotics after smoothing removal. This provides a new analytic instance of large secondary terms in L-function moments and connects to the broader conjectural framework for moments of L-functions, particularly the appearance of secondary terms in Dirichlet L-function moments. The results highlight how explicit-formula methods can reveal subtle secondary-term phenomena beyond the leading order, with potential implications for understanding moments across families of L-functions and their connections to random-matrix theory predictions.

Abstract

In this paper, we discover a secondary term in the asymptotic formula for the mean value of Hecke--Maass special $L$-values $ L (1/2+it_f, f) $ with the average over $f (z)$ in an orthonormal basis of (even or odd) Hecke--Maass cusp forms of Laplace eigenvalue $1/4 + t_f^2$ ($t_f > 0$). To be explicit, we prove $$ \sum_{t_f \leqslant T} ω_f L (1/2+it_f, f) = \frac {T^2} {π^2} + \frac {8T^{3/2}} {3π^{3/2} } + O \big(T^{1+\varepsilon}\big), $$ for any $\varepsilon > 0$, where $ω_f$ are the harmonic weights. This provides a new instance of (large) secondary terms in the moments of $L$-functions -- it was known previously only for the smoothed cubic moment of quadratic Dirichlet $L$-functions. The proof relies on an explicit formula for the smoothed mean value of $L (1/2+it_f, f)$.

Theorems & Definitions (34)

• Theorem 1.1
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof