Shifted convolution of cusp-forms with θ-series

摘要

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.