On higher moments of Dirichlet coefficients attached to symmetric power \(L\)-functions over certain sequences of positive integers (Q6542799)

Let \(j\) be a fixed integer such that \(3\leq j \leq 8\). Let \(f\) be a normalized primitive holomorphic cusp form of even integral weight for the full modular group \(\Gamma =\mathrm{SL}(2, \mathbb{Z})\). Denote by \(\lambda_{\mathrm{sym}^2 f}(n)\) the \(n\)-th normalized coefficient of the Dirichlet expansion of the symmetric square \(L\)-function \(L(\mathrm{sym}^2f, s)\) attached to \(f\).\N\NIn this paper under review, the author is interested in the average behavior of the summatory function \N\[\NS^*_{f,j}(x)=\sum_{\substack{a_1^2+a_2^2+a_3^2+a_4^2+a_4^2+a_5^2+a_6^2\leq x\\ (a_1,\, a_2,\, a_3,\, a_4,\, a_4,\, a_5,\, a_6)\in \mathbb{Z}^6}} \lambda^j_{\mathrm{sym}^2 f}\left(a_1^2+a_2^2+a_3^2+a_4^2+a_4^2+a_5^2+a_6^2\right)\N\]\Nand he shows that \N\[\NS^*_{f,j}(x)=x^3P^*_j(\log x) + O(x^{\theta_j +\varepsilon}), \N\]\Nwhere \(P^*_j(x)\) is a polynomial and \(\theta_j^*\) is an explicit rational number.\N\NIn a similar manner, the author also considers the mean square of coefficients of the Dirichlet expansions of two symmetric power \(L\)-functions attached to two distinct primitive holomorphic cusp forms \(f\in H_{k_1}^*\) and \(g\in H_{k_2}^*\) \N\[\NS_{f, g, i ,j}(x)= \sum_{\substack{a_1^2+a_2^2+a_3^2+a_4^2+a_4^2+a_5^2+a_6^2\leq x\\ (a_1,\, a_2,\, a_3,\, a_4,\, a_4,\, a_5,\, a_6)\in \mathbb{Z}^6}} \lambda^2_{\mathrm{sym}^i f}\left(\sum_{r=1}^6 a_r^2\right)\lambda^2_{\mathrm{sym}^j g}\left(\sum_{r=1}^6 a_r^2\right),\N\]\Nwhere \(i\), \(j \geq 2\) are two fixed positive integers. For any \(\varepsilon>0\), it is shown that \[\NS_{f, g, i ,j}(x)=c_{f, g, i ,j}\, x^3+ O\left( x^{3-\frac{2}{(i+1)^2(j+1)^2}+\varepsilon}\right),\]\Nwhere \(c_{f,g,i,j}\) is an effective constant.