  Upper bound estimates for the exponential sum $$ \sum_{K<\kappa_j\le K'<2K}
\alpha_j H_j^3(1/2) \cos(\k_j\log({4{\rm e}T\over \kappa_j})) \qquad(T^\epsilon
\le K \le T^{1/2-\epsilon}) $$ 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 $\lambda_j =
\kappa_j^2 + {1\over4}$ 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 $H_j(\hf)$ or $H_j^2(\hf)$ replacing $H_j^3(1/2)$ are also considered. The
above sum is conjectured to be $\ll_\epsilon K^{3/2+\epsilon}$, which is proved
to be true in the mean square sense.
