Shifted convolution sums of $G{L}_3$ cusp forms with $\hat{I}¸$-series

Abstract

Let $A_f(1,n)$ be the normalized Fourier coefficients of a Hecke-Maass cusp form $f$ for $S{L}_3(\mathbb{Z})$ and $$r_3(n)=\#\left\{(n_1,n_2,n_3)\in {\mathbb{Z}}^3:n_1^2+n_2^2+n_3^2=n\right\}.$$ Let $1\le h\le X$ and $\ddot{I}•(x)$ be a smooth function compactly supported on $[1/2,1]$. It is shown that $$\sum _{n\ge 1}{A}_f(1,n+h)r_3(n)\ddot{I}•\left(\frac{n}{X}\right){\ll }_{f,\epsilon }{X}^{\frac{3}{2}-\frac{1}{8}+\epsilon }$$ uniformly with respect to the shift $h$.

24 pages. Comments are welcome, Int Math Res Notices (2016)