Czechoslovak Mathematical Journal, Vol. 72, No. 4, pp. 1055-1064, 2022
On higher moments of Hecke eigenvalues attached to cusp forms
Guodong Hua
Received September 10, 2021. Published online June 17, 2022.
Abstract: Let $f$, $g$ and $h$ be three distinct primitive holomorphic cusp forms of even integral weights $k_1$, $k_2$ and $k_3$ for the full modular group $\Gamma={\rm SL}(2,\mathbb{Z})$, respectively, and let $\lambda_f(n)$, $\lambda_g(n)$ and $\lambda_h(n)$ denote the $n$th normalized Fourier coefficients of $f$, $g$ and $h$, respectively. We consider the cancellations of sums related to arithmetic functions $\lambda_g(n)$, $\lambda_h(n)$ twisted by $\lambda_f(n)$ and establish the following results: \sum_{n\leq x}\lambda_f(n)\lambda_g(n)^i\lambda_h(n)^j \ll_{f,g,h,\varepsilon} x^{1- 1/2^{i+j} +\varepsilon} for any $\varepsilon>0$, where $1\leq i\leq2$, $j\geq5$ are any fixed positive integers.
Affiliations: Guodong Hua, School of Mathematics and Statistics, Weinan Normal University, Chaoyang Street, Shaanxi, Weinan 714099, P. R. China, e-mail: gdhua@mail.sdu.edu.cn