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arXiv · 2210.11040

Shifted Convolution Sum for $GL(3) \times GL(2)$ with Weighted Average

Mohd Harun·Saurabh Kumar Singh
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Abstract

In this paper, we will prove the non-trivial bound for the weighted average version of shifted convolution sum for $GL(3)\times GL(2)$, i.e. for any $ε>0$ and $X^{1/4+δ} \leq H \leq X$ with $δ>0$, \[ \frac{1}{H}\sum_{h=1}^\infty λ_f(h) V\left( \frac{h}{H}\right)\sum_{n=1}^\infty λ_π(1,n) λ_g (n+h) W\left( \frac{n}{X} \right)\ll X^{1-δ+ε} \] where $V,W$ are smooth compactly supported funtions, $λ_f(n), λ_g(n)$ and $λ_π(1,n)$ are the normalized n-th Fourier coefficients of $SL(2,\mathbb{Z})$ Hecke-Maass cusp forms $f,g$ and $SL(3,\mathbb{Z})$ Hecke-Maass cusp form $π$, respectively.

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Mohd Harun, Saurabh Kumar Singh. 2023-11-13. Shifted Convolution Sum for $GL(3) \times GL(2)$ with Weighted Average. https://arxiv.org/abs/2210.11040

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