On the Fourth Power Moment of Fourier Coefficients of Cusp Form

Abstract

Let $a(n)$ be the Fourier coefficients of a holomorphic cusp form of weight $\kappa =2n⩾12$ for the full modular group and $A(x)=\sum _{n⩽x}a(n)$. 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}{64κπ^4}s_{4;2}(\tilde{a}) T^{2κ}+O\big(T^{2κ-δ_4+\varepsilon}\big) \end{equation*} with ${\delta }_4=1/8$, which improves the previous result.

11 pages