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 ≥ 2 is any integer. Let L (s,symj 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 SL2 (Z). Also, let λsym j f (n) be the n-th normalized Dirichlet coefficient of L (s,symj f). In this paper, we establish asymptotic formulas for sums of Dirichlet coefficients λsym j f (n) over two sparse sequences of positive integers, which improves previous results.