Explicit count of integral ideals of an imaginary quadratic field

Abstract

We provide explicit bounds for the number of integral ideals of norms at most X in \(\mathbb {Q}[\sqrt{d}]\) when \(d <0\) is a fundamental discriminant with an error term of size \(\mathcal {O}(X^{1/3})\). In particular, we prove that, when \(\chi \) is the non-principal character modulo 3 and \(X\ge 1\), we have \(\sum _{n\le X}(1\!\!\!1\star \chi )(n) = \frac{\pi X}{3\sqrt{3}} +\mathcal {O}^*(1.94\,X^{1/3})\), and that, when \(\chi \) is the non-principal character modulo 4 and \(X\ge 1\), we have \(\sum _{n\le X}(1\!\!\!1\star \chi )(n) = \frac{\pi X}{4} +\mathcal {O}^*(1.4\,X^{1/3})\). Résumé. Nous dénombrons de façon explicite avec un terme d’erreur \(\mathcal {O}(X^{1/3})\) le nombre d’idéaux entiers de norme au plus X du corps \(\mathbb {Q}[\sqrt{d}]\) lorsque \(d <0\) est un discriminant fondamental. Nous montrons en particulier que, lorsque \(\chi \) est le caractère non principal modulo 3 et \(X\ge 1\), nous avons \(\sum _{n\le X}(1\!\!\!1\star \chi )(n) = \frac{\pi X}{3\sqrt{3}} +\mathcal {O}^*(1.94\,X^{1/3})\), et que , lorsque \(\chi \) est le caractère non principal modulo 4 et \(X\ge 1\), nous avons \(\sum _{n\le X}(1\!\!\!1\star \chi )(n) = \frac{\pi X}{4} +\mathcal {O}^*(1.4\,X^{1/3} )\).