On the moments of Hecke series at central points. II

Abstract

We prove, in standard notation from spectral theory, the asymptotic formula $(B > 0)$ $$\sum_{\kappa_{j}\leq T} \alpha_{j} H_{j}(\frac{1}{2}) = \left(\frac{T}{\pi}\right)^2 - BT \log T + O(T(\log T)^{1/2}),$$ by using an approximate functional equation for $H_{j}(\frac{1}{2})$ and the Bruggeman-Kuznetsov trace formula. We indicate how the error term may be improved to $O(T(\log T)^{\varepsilon})$.