On exponential sums with Hecke series at central points

Abstract

Upper bound estimates for the exponential sum $$ \sum_{K<κ_j\le K'<2K} α_j H_j^3(1/2) \cos(\k_j\log({4{\rm e}T\over κ_j})) \qquad(T^ε\le K \le T^{1/2-ε}) $$ 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 ${\lambda }_j={\kappa }_j^2+\frac{1}{4}$ 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 $H_j(\hf)$ 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.

31 pages