Sum of the $GL(3)$ Fourier coefficients over quadratics

Abstract

Let $A(n)$ denote the $(1,n)\text{-th}$ Fourier coefficient of a $SL(3,\mathbb{Z})$ Hecke eigenform or the ternary divisor function $d_3(n)$. Let $Q(x,y)$ be a symmetric positive definite quadratic form. This article establishes an asymptotic formula with a power-saving error term for the following sum \begin{equation*} \sum_{1 \leqslant m \leqslant X} \sum_{1 \leqslant n\leqslant Y} A(Q(m,n)), \end{equation*} where $X>1$ and $Y⩽X$.

This version is the published version of the article that appeared in The Ramanujan Journal

References in corpus (2)

• Moments and hybrid subconvexity for symmetric-square L-functions
• Sum of the $GL(3)$ Fourier coefficients over mixed powers