On $\mathrm{G}{\mathrm{L}}_3$ Fourier coefficients over values of mixed powers

Abstract

Let $A_{\ddot{I}}€(n,1)$ be the $(n,1)$-th Fourier coefficient of the Hecke-Maass cusp form $\ddot{I}€$ for $\mathrm{S}{\mathrm{L}}_3(\mathbb{Z})$ and $\ddot{I}‰(x)$ be a smooth compactly supported function. In this paper, we prove a nontrivial upper bound for the sum $$\sum _{\genfrac{}{}{0ex}{}{n_1,\cdots ,n_{\ell },n_{\ell +1}\in {\mathbb{Z}}_{+}}{n=n_1^r+\cdots +n_{\ell }^r+n_{\ell +1}^s}}{A}_{\ddot{I}}€(n,1)\ddot{I}‰\left(n/X\right),$$ where $r\ge 2$, $s\ge 2$ and $\ell \ge 2^{r-1}$ are integers.