论文标题

$ gl(3)\ times gl(2)$的转移卷积总和,加权平均值

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

论文作者

Harun, Mohd, Singh, Saurabh Kumar

论文摘要

在本文中,我们将证明$ gl(3)\ times gl(2)$的加权平均版本的非平凡限制,即对于任何$ε> 0 $和$ X^{1/4+δ} \ leq leq H \ leq h \ leq h \ leq x $,带有$Δ> 0 $,$,$,$,$,$,, \ [ \ frac {1} {h} \ sum_ {h = 1}^\inftyλ_f(H) \ right)\ ll x^{1-δ+ε} \] 其中$ v,w $是平滑的紧凑支持的功能,$λ_f(n),λ_g(n)$和$λ_π(1,n)$是$ sl(2,\ m athbb {z})的均标准化n-傅立叶系数(\ mathbb {z})$尖端$π$。

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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