## Experimental Mathematics

### An investigation of bounds for the regulator of quadratic fields

#### 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 $\eps_{0} > 1$, called the fundamental unit of $\K$. The natural logarithm of $\eps_{0}$ 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.

#### Article information

Source
Experiment. Math., Volume 4, Issue 3 (1995), 211-225.

Dates
First available in Project Euclid: 3 September 2003