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 \(j\ge 2\) is any integer. Let \(L(s, \textrm{sym}^j f)\) 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\(_{2}(\mathbb {Z})\). Also, let \(\lambda _{\textrm{sym}^j f}(n)\) be the n-th normalized Dirichlet coefficient of \(L(s, \textrm{sym}^j f)\). In this paper, we establish asymptotic formulas for sums of Dirichlet coefficients \(\lambda _{\textrm{sym}^j f}(n)\) over two sparse sequences of positive integers, which improves previous results.