Sum of the $GL(3)$ Fourier coefficients over mixed powers

Abstract

Let $A(n)$ be the $(1,n)$-th Fourier coefficients of $SL(3,\mathbb{Z})$ Hecke-Maass cusp form, denoted as $A(1,n)$ or the triple divisor function, denoted as $d_3(n)$. Let $k \geqslant3$ be an integer. In this paper, we establish an asymptotic formula for the sum \begin{equation*} \mathop{\sum}_{\substack{1 \leqslant n_1, n_2 \leqslant X^{1/2} \\ 1 \leqslant n_3 \leqslant X^{1/k}}} A(Q(n_1,n_2) + n_3^k)\mathsf{a}(n_3), \end{equation*} where $\mathsf{a}(n)$ is either von-Mangoldt function or identity function, and $Q(x,y) \in \mathbb{Z}[x,y]$ is a binary quadratic polynomial. When $A(n)=A(1,n)$, then $\mathsf{a}(n)$ can be any bounded arithmetical function.