On mean values of some arithmetic functions involving different number fields (Q746389)

The authors of the paper investigate the average asymptotic behaviour of the sum \[ \sum_{n\leq x}\tau_{k_1}^{K_1}(n)\tau_{k_2}^{K_2}(n)\dots \tau_{k_l}^{K_l}(n), \] where \(l\geq1, k_1,k_2,\dots, k_l\geq 2\) are integers, \(K_1,K_2,\dots, K_l\) are the number fields of finite degrees over the rational field \(\mathbb{Q}\) and \(\tau_k^K\) denote the \(k\)-dimensional divisor function in the number field \(K\). For instance, the authors of the paper prove that \[ \sum_{n\leq x}\tau_{k_1}^{K_1}(n)\tau_{k_2}^{K_2}(n)\dots \tau_{k_l}^{K_l}(n)\ll x(\log x)^{\displaystyle\prod_{j=1}^{l}k_j\prod_{j=1}^{l-1}d_j-1} \] if the degrees \(d_1,d_2,\dots,d_l\) of the number fields \(K_1,K_2,\dots, K_l\) satisfy the condition \(d_1\leq d_2 \leq\dots \leq d_l\).