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

Abstract

In this paper, we discover a secondary term in the asymptotic formula for the mean value of Hecke--Maass special $L$-values $L(1/2+i{t}_f,f)$ with the average over $f(z)$ in an orthonormal basis of Hecke--Maass cusp forms of Laplace eigenvalue $1/4+t_f^2$ ($t_f>0$). To be explicit, we prove $$\sum _{t_f⩽T}{\omega }_{f}L(1/2+i{t}_f,f)=\frac{T^2}{{\pi }^2}+\frac{4T^{3/2}}{3{\pi }^{3/2}}+O\left(T^{1+\epsilon }\right),$$ for any $\epsilon >0$, 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 $L(1/2+i{t}_f,f)$.

26 pages; some errors corrected (no secondary term for odd cusp forms); to appear in Pacific J. Math