On $\rm GL_3$ Fourier coefficients over values of mixed powers

Abstract

Let $A_π(n,1)$ be the $(n,1)$-th Fourier coefficient of the Hecke-Maass cusp form $π$ for $\rm SL_3(\mathbb{Z})$ and $ ω(x)$ be a smooth compactly supported function. In this paper, we prove a nontrivial upper bound for the sum $$ \sum_{n_1,\cdots,n_\ell,n_{\ell+1}\in \mathbb{Z}_+ \atop n=n_1^r+\cdots+n_{\ell}^r+n_{\ell+1}^s} A_π(n,1)ω\left(n/X\right), $$ where $r\geq2$, $s\geq 2$ and $\ell\geq 2^{r-1}$ are integers.