On the ideal theorem for number fields

Abstract

Let $K$ be an algebraic number field and $\nu_K$ be the ideal-counting function of $K$. Many authors have estimated the remainder term $\Delta_n(x,K)$ in the asymptotic formula of the average order of $\nu_K$. The purpose of this work is twofold: we first generalize Müller's method to the $n$-dimensional case and improve on Nowak's result. A key part in the proof is played by a~profound result on a triple exponential sum recently derived by Robert \& Sargos.