On sums of Hecke series in short itervals

Abstract

We prove $\sum_{K-G}\le κ_j \le K+G}α_j H_j^3(1/2) \ll_εGK^{1+ε}$ 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 $H_j(1/2){\ll }_{\epsilon }{\kappa }_j^{1/3+\epsilon }$.

16 pages