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.

假设x是一个足够大的数，并且\(j\ge 2\)是任意整数。令\(L(s, \textrm{sym}^jf)\)为与全模群 SL \(_{2}的权重k的原始全纯尖点形式f相关联的第j个对称幂L函数(\mathbb {Z})\)。另外，令\(\lambda _{\textrm{sym}^jf}(n)\)为\(L(s, \textrm{sym}^jf)\)的第n个归一化狄利克雷系数。在本文中，我们建立了两个正整数稀疏序列的狄利克雷系数和\(\lambda _{\textrm{sym}^jf}(n)\)的渐近公式，这改进了之前的结果。