A positive density of fundamental discriminants with large regulator
Let \(\varepsilon_D\) be the fundamental unit of a quadratic number field of discriminant \(D\). The authors prove under some mild technical restrictions that \[ \sharp \{D\leq X : \varepsilon_D\geq D^{3-\delta}\}\gg_\delta X \] for any \(\delta>0\), provided the norm of \(\varepsilon_D\) is positive. Moreover, they show that there exists a constant \(\delta>0\) such that \[ \sum_{D\leq x;\; 4\| D}h(D)\leq \left(\frac{8}{21 \pi^2}C_0-\delta\right) \frac{x^{3/2}}{\log x}, \] where \(C_0\) is defined by a converging Euler product.
scientific article; zbMATH DE number 6177098
scientific article