Let \(\lambda _\pi (1,\ldots ,1,n)\) be the normalized Fourier coefficients of an even Hecke–Maass form \(\pi \) for \(SL(m, {\mathbb {Z}})\) with \(m\ge 3\), and \(r_{3}(n)=\#\{(n_1,n_2,n_3)\in {\mathbb {Z}}^3:n=n_1^2+n_2^2+n_3^2\}\). In this paper, we introduce a refined version of the circle method to derive a sharp bound for the shifted convolution sum of GL(m) Fourier coefficients \(\lambda _\pi (1,\ldots ,1 ,n)\) and \(r_{3}(n)\), which improves previous results (even under the generalized Ramanujan conjecture).

中文翻译：

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GL(m) 尖点形式的移位卷积和 $$\Theta $$ Θ -series

让\（\拉姆达_ \ PI（1，\ ldots，1，N）\）是偶数赫克-Maass的形式的归一化的傅立叶系数\（\ PI \）为\（SL（M，{\ mathbb {Z }})\)与\(m\ge 3\)和\(r_{3}(n)=\#\{(n_1,n_2,n_3)\in {\mathbb {Z}}^3:n =n_1^2+n_2^2+n_3^2\}\)。在本文中，我们引入了圆形方法的改进版本，以推导出GL ( m ) 傅立叶系数\(\lambda _\pi (1,\ldots ,1 ,n)\)和\(r_{3}(n)\)，这改善了以前的结果（即使在广义拉马努金猜想下）。