Counting ideals in ray classes (Q2109396)

Let \(I\) stand for the monoid of the integral ideals of a number field \(k\) of degree \(n:=[k:\mathbb Q]\), let \(B\) be a ray class of \(k\), and let \[ \mathcal{N} (B, x):= \{{\mathfrak a}\mid {\mathfrak a}\in B\cap I, N{\mathfrak a}\leq x\}. \] The authors prove that \[ |\mathcal{N} (B, x) - \alpha (B) x| \leq x^{(n-1)/n}f(B) + g(B)\text{ for }x\geq 1, \] where \(\alpha (B), f(B)\), and \(g(B)\) are expressed in terms of \(k\) and \(B\). According to the authors, they follow the approach developed by \textit{K. Debaene} [Int. J. Number Theory 15, No. 5, 883--905 (2019; Zbl 1456.11218)]. To obtain their asymptotic formula, the authors employ technique of geometry of numbers, citing, in particular, a recent theorem of \textit{M. Widmer} [Trans. Am. Math. Soc. 362, No. 9, 4793--4829 (2010; Zbl 1270.11064)] on the number of lattice points in a bounded subset of \(\mathbb R^{m}, m\geq 2\), of Lipschitz class.