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arXiv · 1702.00163

On the Fourth Power Moment of Fourier Coefficients of Cusp Form

Jinjiang Li·Panwang Wang·Min Zhang
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Abstract

Let $a(n)$ be the Fourier coefficients of a holomorphic cusp form of weight $\kappa=2n\geqslant12$ for the full modular group and $A(x)=\sum\limits_{n\leqslant x}a(n)$. In this paper, we establish an asymptotic formula of the fourth power moment of $A(x)$ and prove that \begin{equation*} \int_1^TA^4(x)\mathrm{d}x=\frac{3}{64\kappa\pi^4}s_{4;2}(\tilde{a}) T^{2\kappa}+O\big(T^{2\kappa-\delta_4+\varepsilon}\big) \end{equation*} with $\delta_4=1/8$, which improves the previous result.

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Jinjiang Li, Panwang Wang, Min Zhang. 2017-02-01. On the Fourth Power Moment of Fourier Coefficients of Cusp Form. https://doi.org/10.1007/s10114-017-6508-5

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