On general divisor problems of Hecke eigenvalues of cusp forms (Q6061181)

Let \(f\) be a primitive holomorphic cusp form of even integral weight \(k\) for the full modular group \(\Gamma = SL(2,\mathbb{Z})\). Then we have \[ f(z) = \sum_{n=1}^{\infty} \lambda_f(n)n^{(k-1)/2}e(nz), \quad \text{Im}(z) > 0, \] where \(e(z) = e^{2niz}\), and \(\lambda_f(n)\) are the normalized eigenvalues of the Hecke operators \(T_n\). \par The author considers the average behaviour of the sum \[S_{j_1,\dots,j_r}(f;x) := \sum_{n\leq x} \lambda_{f,j_1,\dots,j_r}(n), \] where \[\lambda_{f,j_1,\dots,j_r}(n) := \sum_{n=n_1 \cdots n_r} \lambda_f(n_1)^{j_1} \cdots \lambda_f(n_r)^{j_r}, \] with \(j_1,\dots,j_r \geq 1\) and \(r\geq 2\). He obtains asymptotic formula for \(S_{j_1,\dots,j_r}(f;x)\) (Theorem 1.1), generalizing the result of \textit{W. Zhang} [Ramanujan J. 53, No. 1, 75--83 (2020; Zbl 1469.11084)]. The proof uses general results of \textit{Y.-K. Lau} and \textit{G. Lü} [Q. J. Math. 62, No. 3, 687--716 (2011; Zbl 1269.11044)]. Similar arguments give asymptotic formula for \(S^*_{j_1,\dots,j_r}(f;x)\) (Theorem 1.2), where \[ S^*_{j_1,\dots,j_r}(f;x) := \sum_{n\leq x} \lambda^*_{f,j_1,\dots,j_r}(n), \] and the definition of \(\lambda^*_{f,j_1,\dots,j_r}(n)\) is similar to that of \(\lambda_{f,j_1,\dots,j_r}(n)\), but with additional conditions \(n_1 = a_1^2+b_1^2\),\dots, \(n_r = a_r^2+b_r^2\). \par Theorems 1.1 and 1.2 generalize previous results obtained by many authors.