The average behaviour of a hybrid arithmetic function associated to cusp form coefficients over certain sparse sequence

Abstract

Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma=SL(2,\mathbb{Z})$, and let $\lambda_{f}(n),$ $\sigma(n)$ and $\varphi(n)$ be the $n$th normalized Fourier coefficient of the cusp form $f$, the sum-of-divisors function and the Euler totient function, respectively. In this paper, we investigate the asymptotic behaviour of the following summatory function
\begin{eqnarray} \nonumber
S_{j,b,c}(x) := \sum_{\substack{n=a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2}\leq x \\ (a_{1},a_{2},a_{3},a_{4})\in\mathbb{Z}^{4}}}\lambda_{f}^{j}(n)\sigma^{b}(n)\varphi^{c}(n),
\end{eqnarray}
where $j\geq 2$ is any given integer. In a similar manner, we also establish other similar results related to normalized coefficients of the symmetric power $L$-functions associated to holomorphic cusp form $f$.