Generalized divisor problem associated with the number of integral ideals

In this article, we study the power moments of the function τ k , K ( n ) , which counts the number of representations of n as a product of norms of k ideals in the ring of integers of a non-normal cubic field K . Additionally, we analyze the statistical properties of τ k , K ( n ) , including its mean and variance. Furthermore, we establish asymptotic formulae for cases where n is represented by a fixed mixed form with positive integral coefficients.