Date: 12 Jan 2011

Shifted convolution of cusp-forms with θ-series

Abstract

We generalize the classical Voronoi formula for $$r_{l}(n) = \#\{ (n_{1}, \ldots , n_{l}) \in \mathbf{Z}^{l}, n_{1}^{2} + \cdots + n_{l}^{2} = n \},$$ and as an application, we derive a sharp bound for the shifted convolution sum convolving the Fourier coefficients of holomorphic cusp forms with those of theta series.

Communicated by U. Kühn.
The research is partially supported by NSF grant DMS-0855600.