Residue classes modulo a prime number in an algebraic number field (Q1113940)

Let K be a finite degree n algebraic extension of the rational number field and p a prime number. Denote by \(N_ p(A)=\min_{\alpha \in A}| Nm(\alpha)|, \) where A is a residue class in the ring of integers of the field K modulo the ideal (p). Consider the mean \(M(K,p)=\frac{1}{p^ n}\sum_{A(mod(p))}N_ p(A), \) where the sum extends to all the residue classes modulo (p). It is proved that if K has one fundamental unit then (i) for arbitrary prime numbers p the estimate \(M(K,p)=O(p^ n(\ln (p))^{-1/3}) \) holds, (ii) for almost all prime numbers p in the sense of asymptotic density the estimate \(M(K,p)=O(Nm(p)^{1-1/6n}(\ln (p))^{1/3}) \) holds, (iii) for some infinite sequence of integer ideals \({\mathfrak q}\) of K the estimate \[ \frac{1}{Nm({\mathfrak q})}\sum_{B(mod {\mathfrak q})}\min_{\beta \in B}| Nm(\beta)| =O(Nm({\mathfrak q})^{1- 1/6n}\ln^{1/3}Nm({\mathfrak q})) \] holds, when B are residue classes in the ring of integers of the field K modulo the ideal \({\mathfrak q}\).