We produce nontrivial asymptotic estimates for shifted sums of the form $\sum a(h)b(m)c(2m-h)$, in which $a(n),b(n),c(n)$ are un-normalized Fourier coefficients of holomorphic cusp forms. These results are unconditional, but we demonstrate how to strengthen them under the Riemann Hypothesis. As an application, we show that there are infinitely many three term arithmetic progressions $n-h, n, n+h$ such that $a(n-h)a(n)a(n+h) \neq 0$.

中文翻译：

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尖峰形式系数的三重相关和

我们为 $\sum a(h)b(m)c(2m-h)$ 形式的移位和产生非平凡渐近估计，其中 $a(n),b(n),c(n)$ 是 un -全纯尖端形式的归一化傅立叶系数。这些结果是无条件的，但我们展示了如何在黎曼假设下加强它们。作为一个应用，我们证明存在无限多个三项等差数列 $nh, n, n+h$ 使得 $a(nh)a(n)a(n+h) \neq 0$。