The Average Behaviors of the Fourier Coefficients of j-th Symmetric Power L-Function over Two Sparse Sequences of Positive Integers

Suppose that x is a sufficiently large number and \(j\ge 2\) $j\ge 2$ is any integer. Let \(L(s, \textrm{sym}^j f)\) $L(s,{\text{sym}}^{j}f)$ be the j-th symmetric power L-function associated with the primitive holomorphic cusp form f of weight k for the full modular group SL \(_{2}(\mathbb {Z})\) ${}_2(\mathbb{Z})$ . Also, let \(\lambda _{\textrm{sym}^j f}(n)\) ${\lambda }_{{\text{sym}}^{j}f}(n)$ be the n-th normalized Dirichlet coefficient of \(L(s, \textrm{sym}^j f)\) $L(s,{\text{sym}}^{j}f)$ . In this paper, we establish asymptotic formulas for sums of Dirichlet coefficients \(\lambda _{\textrm{sym}^j f}(n)\) ${\lambda }_{{\text{sym}}^{j}f}(n)$ over two sparse sequences of positive integers, which improves previous results.