Higher moment of coefficients of Dedekind zeta function (Q2184261)

Let \(\zeta_K(s) = \sum_{n=1}^\infty\frac{a_K(n)}{n^s}\) be the Dedekind zeta-function of a non-normal cubic field \(K\) and let \[ S_{K,m}(x) = \sum_{n_1^2+n_2^2\le x}a_K^m(n_1^2+n_2^2). \] The equality \[ S_{K,1}(x) = xP(\log x) +O_\varepsilon\left(x^{3/5+\varepsilon}\right), \] where \(P(X)\) is a quadratic polynomial, has been established by \textit{Z. Yang} [Front. Math. China 12, No. 4, 981--992 (2017; Zbl 1427.11122)]. The author shows that for \(2\le m\le 8\) one has \[ S_{K,m}(x) = xP_m(\log x) +O_\varepsilon\left(x^{\theta_m+\varepsilon}\right), \] where \(P_m(X)\) is a polynomial of degree \(\eta_m\), the numbers \(\theta_m\) and \(\eta_m\) given explicitly. For example, one has \(\eta_2=1, \theta_2=51/59\) and \(\eta_8=609, \theta_8=18356/18359\).