A shifted convolution sum for $GL(3)\times GL(2)$

Abstract

In this paper, we estimate the shifted convolution sum \[\sum_{n\geqslant1}λ_1(1,n)λ_2(n+h)V\Big(\frac{n}{X}\Big),\] where $V$ is a smooth function with support in $[1,2]$, $1⩽|h|⩽X$, ${\lambda }_1(1,n)$ and ${\lambda }_2(n)$ are the $n$-th Fourier coefficients of $SL(3,\mathbf{Z})$ and $SL(2,\mathbf{Z})$ Hecke-Maass cusp forms, respectively. We prove an upper bound $O(X^{\frac{21}{22}+\epsilon })$, updating a recent result of Munshi.

18 pages. All comments are welcome!