# On the Fourth Power Moment of Fourier Coefficients of Cusp Form

## TL;DR

This paper derives an asymptotic formula for the fourth power moment of the partial sums of Fourier coefficients of holomorphic cusp forms, improving previous bounds and deepening understanding of their distribution.

## Contribution

It establishes a refined asymptotic formula for the fourth power moment of Fourier coefficient sums, enhancing previous results with a better error term.

## Key findings

- Derived an asymptotic formula for the fourth power moment of A(x)
- Improved the error term in the asymptotic estimate
- Enhanced understanding of the distribution of Fourier coefficients

## 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.