On sums of Hecke series in short itervals

Abstract

We prove $\sum_{K-G}\le \kappa_j \le K+G}\alpha_j H_j^3(1/2) \ll_\epsilon GK^{1+\epsilon}$ for $K^\epsilon \le G \le K$, where $\alpha_j = |\rho_j(1)|^2(\cosh \pi\kappa_j)^{-1}$, and $\rho_j(1)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue $\lambda_j = \kappa_j^2 + 1/4$ to which the Hecke series $H_j(s)$ is attached. This result yields the new bound $H_j(1/2) \ll_\epsilon \kappa_j^{1/3+\epsilon}$.