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arXiv · 2303.15856

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

Himanshi Chanana·Saurabh Kumar Singh
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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 $Y\leqslant X$.

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Himanshi Chanana, Saurabh Kumar Singh. 2023-03-28. Sum of the $GL(3)$ Fourier coefficients over quadratics. https://doi.org/10.1007/s11139-026-01362-0

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