Oscillations of Fourier coefficients over the sparse set of integers

Lalit Vaishya
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Abstract

Let \(f \in S_{k}(\Gamma _{0}(N))\) be a normalized Hecke eigenforms of integral weight k and level \(N \ge 1\). In the article, we establish the asymptotics of power moment associated to the sequences \(\{\lambda _{f \otimes f \otimes f}(\mathcal {Q}(\underline{x}))\}_{\mathcal {Q} \in \mathcal {S}_{D}, \underline{x} \in \mathbb {Z}^{2}}\) and \(\{\lambda _{f \otimes \mathrm{sym^{2}}f}(\mathcal {Q}(\underline{x}))\}_{\mathcal {Q} \in \mathcal {S}_{D}, \underline{x} \in \mathbb {Z}^{2}}\) where \(\mathcal {S}_{D}\) denotes the set of inequivalent primitive integral positive-definite binary quadratic forms (reduced forms) of fixed discriminant \(D < 0.\) As a consequence, we prove results concerning the behaviour of sign changes associated to these sequences.

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稀疏整数集上的傅立叶系数振荡
设 \(f \in S_{k}(\Gamma _{0}(N))\) 是积分权重为 k 且级别为 \(N \ge 1\) 的归一化 Hecke 特征形式。在这篇文章中,我们建立了与序列 \(\{lambda _{f \otimes f \otimes f}(\mathcal {Q}(\underline{x}))\}_{\mathcal {Q}) 相关的幂矩渐近论。\在 \mathcal {S}_{D}, (下划线{x})中\)和(({lambda _{f times \mathrm{sym^{2}}f}(\mathcal {Q}(\underline{x}))\}_{\mathcal {Q}}\in \mathcal {S}_{D}, \underline{x}\其中 \(\mathcal {S}_{D}\) 表示固定判别式 \(D < 0.\) 的不等价原始积分正定二元二次形式(简化形式)的集合。 因此,我们证明了与这些序列相关的符号变化行为的结果。
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