An Investigation of Bounds for the Regulator of Quadratic Fields

Michael J. Jacobson, Jr. Richard F. Lukes Hugh C. Williams

Abstract

It is well known that the nontorsion part of the unit group of a real quadratic field K is cyclic. With no loss of generality we may assume that it has a generator ϵ_{0} > 1, called the fundamental unit of K. The natural logarithm of ϵ_{0} is called the regulator R of K. This paper considers the following problems: How large, and how small, can R get? And how often?

The answer is simple for the problem of how small R can be, but seems to be extremely difficult for the question of how large R can get. In order to investigate this, we conducted several large-scale numerical experiments, involving the Extended Riemann Hypothesis and the Cohen–Lenstra class number heuristics. These experiments provide numerical confirmation for what is currently believed about the magnitude of R.