Secondary term for the mean value of Maass special $L$-values

Zhi Qi

Vol. 345 (2026), No. 2, 385–416

DOI: 10.2140/pjm.2026.345.385

Abstract

We discover a secondary term in the asymptotic formula for the mean value of Hecke–Maass special $L$-values $L\left(\frac{1}{2}+i{t}_f,f\right)$ with the average over $f(z)$ in an orthonormal basis of Hecke–Maass cusp forms of Laplace eigenvalue $\frac{1}{4}+t_f^2$ ($t_f>0$). To be explicit, we prove

$$\sum _{t_f\le T}{\omega }_{f}L\left(\frac{1}{2}+i{t}_f,f\right)=\frac{T^2}{{\pi }^2}+\frac{4T^{3∕2}}{3{\pi }^{3∕2}}+O(T^{1+𝜀})$$

for any $𝜀>0$, where ${\omega }_f$ 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 $L\left(\frac{1}{2}+i{t}_f,f\right)$.

Keywords

Maass forms, $L$-functions, Kuznetsov formula

Mathematical Subject Classification

Primary: 11F72, 11M41

Milestones

Received: 19 September 2025

Revised: 31 July 2026

Accepted: 4 August 2026

Published: 26 October 2026