On exponential sums with Hecke series at central points

Abstract

Upper bound estimates for the exponential sum $$\sum_{K<\kappa_j\le K^\prime<2K} \alpha_j H_j^3(\tfrac{1}{2}) \cos(\kappa_j\log(\frac{4eT}{\kappa_j}))\qquad(T^\varepsilon \le K \le T^{1/2-\varepsilon})$$ are considered, 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 $\lambda_j = \kappa_j^2 + \tfrac{1}{4}$ to which the Hecke series $H_j(s)$ is attached. The problem is transformed to the estimation of a classical exponential sum involving the binary additive divisor problem. The analogous exponential sums with $H_j(\tfrac{1}{2})$ or $H_j^2(\tfrac{1}{2})$ replacing ${H_j^3(\tfrac{1}{2})}$ are also considered. The above sum is conjectured to be $\ll_\varepsilon K^{3/2+\varepsilon}$, which is proved to be true in the mean square sense.