Shifted convolution sums involving theta series

Abstract

Let $f$ be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus and denote by $\hat{I}{»}_f(n)$ its $n$-th Hecke eigenvalue. Let $$r(n)=\#\left\{(n_1,n_2)\in {\mathbb{Z}}^2:n_1^2+n_2^2=n\right\}.$$ In this paper, we study the shifted convolution sum $${\mathcal{S}}_h(X)=\sum _{n\le X}\hat{I}{»}_f(n+h)r(n),1\le h\le X,$$ and establish uniform bounds with respect to the shift $h$ for ${\mathcal{S}}_h(X)$.

21 pages. Comments are welcome! To appear in Ramanujan J