Summary: Suppose that \(E\) is an algebraic number field over the rational field \(\mathbb{Q}\). Let \(a\left(n \right)\) be the number of integral ideals in \(E\) with norm \(n\). Let also \(\Delta \left(x \right)\) denote the remainder term in the asymptotic formula for the average behavior \({\sum_{n \leq x}}{\left({a\left({{n^2}} \right)} \right)^l}\). In this paper, the sharp bound for \[ \int_1^X {{\Delta ^2}\left(x \right){\text{dx}}} \] is given by analytical method.