Sum of the triple divisor function and Fourier coefficients of $SL(3,\mathbb{Z})$ Hecke-Maass forms over quadratics

Abstract

Let $\mathcal{A}(n)$ be the $(1,n)-th$ Fourier coefficients of $SL(3,\mathbb{Z})$ Hecke-Maass cusp form i.e. $\Lambda(1,n)$ or the triple divisor function $d_3(n)$, which is the number of solutions of the equation $r_1r_2r_3 = n$ with $r_1, r_2, r_3 \in \mathbb{Z}^+.$ We establish estimates for \begin{equation*} \sum_{1 \leq n_1,n_2\leq X} \mathcal{A}(Q(n_1,n_2)) \end{equation*} where $Q(x,y) \in \mathbb{Z}[x,y]$ is a symmetric positive definite quadratic form.