In this paper, we will prove the non-trivial bound for the weighted average version of shifted convolution sum for $GL(3)\times GL(2)$, i.e. for any $\epsilon >0$ and $X^{ 1/4+\delta} \leq H \leq X$ with $\delta >0$, \[ \frac{ 1}{ H}\sum_{ h=1}^\infty \lambda_f(h) V\left( \frac{ h}{ H}\right)\sum_{ n=1}^\infty \lambda_{ \pi}(1,n) \lambda_g (n+h) W\left( \frac{ n}{ X} \right)\ll X^{ 1-\delta+\epsilon} \] where $V,W$ are smooth compactly supported funtions, $\lambda_f(n), \lambda_g(n)$ and $\lambda_{ \pi}(1,n)$ are the normalized n-th Fourier coefficients of $SL(2,\mathbb{ Z})$ Hecke-Maass cusp forms $f,g$ and $SL(3,\mathbb{ Z})$ Hecke-Maass cusp form $\pi$, respectively.
Categories: math.NT