On the mean value of the index of composition of an integral ideal (Q627622)

Let \(K\) be an algebraic number field. For an integral ideal \({\mathfrak A}\) the authors define its index of composition as \[ \lambda({\mathfrak A}):=\frac{\log N({\mathfrak A})}{\log \prod_{{\mathfrak P}\mid {\mathfrak A}}N({\mathfrak P})},\gamma({\mathfrak A}):=\prod_{{\mathfrak P}\mid {\mathfrak A}}N({\mathfrak P}) , \] where \(N({\mathfrak A})\) is the norm of \({\mathfrak A}\) and \({\mathfrak P}\) are prime ideals. For an integer \(k\geq 1\) they prove asymptotic formulas for \(\sum_{N({\mathfrak A})\leq x}\lambda^k({\mathfrak A})\) and \(\sum_{N({\mathfrak A})\leq x}\lambda^{-k}({\mathfrak A})\).