On the Regulators of Real Quadratic Number Fields

TL;DR

This paper investigates the properties of quadratic integers with small norms in real quadratic number fields, suggesting that a certain regulator bound R >> (log D)^2 holds for most such fields.

Contribution

It provides evidence and corrections regarding the behavior of regulators in real quadratic fields, emphasizing the typical size of R relative to (log D)^2.

Findings

R is generally much larger than (log D)^2 in most real quadratic fields

The paper corrects previous errors related to quadratic integers and regulators

Most real quadratic fields satisfy the inequality R >> (log D)^2

Abstract

This paper deals with the quadratic integers of small norms and asserts that in some sense R >> (log D)^2 is true for almost all real quadratic number fields. (A few errata is corrected.)