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# Higher Moments of Fourier Coefficients of Cusp Forms

Published:2015-05-21
Printed: Sep 2015
• Guangshi Lü,
School of Mathematics, Shandong University, Jinan, Shandong 250100, China
Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group $SL(2, \mathbb{Z})$. Let $\lambda_f(n)$, $\lambda_g(n)$, $\lambda_h(n)$ be the $n$th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms $f(z) \in S_{k_1}(\Gamma), g(z) \in S_{k_2}(\Gamma), h(z) \in S_{k_3}(\Gamma)$ respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as $\lambda_f(n)^4\lambda_g(n)^2$, $\lambda_g(n)^6$, $\lambda_g(n)^2\lambda_h(n)^4$, and $\lambda_g(n^3)^2$ twisted by the arithmetic function $\lambda_f(n)$.
 Keywords: Fourier coefficients of automorphic forms, Dirichlet series, triple product $L$-function, Perron's formula
 MSC Classifications: 11F30 - Fourier coefficients of automorphic forms 11F66 - Langlands $L$-functions; one variable Dirichlet series and functional equations