## Experimental Mathematics

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

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

Michael J. Jacobson, Jr., Richard F. Lukes, and 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 $\eps_{0} > 1$, 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

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1062621079

**Mathematical Reviews number (MathSciNet)**

MR1387478

**Zentralblatt MATH identifier**

0859.11057

**Subjects**

Primary: 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]

Secondary: 11R11: Quadratic extensions 11R29: Class numbers, class groups, discriminants 11Y40: Algebraic number theory computations

#### Citation

Jacobson, Jr., Michael J.; Lukes, Richard F.; Williams, Hugh C. An investigation of bounds for the regulator of quadratic fields. Experiment. Math. 4 (1995), no. 3, 211--225. https://projecteuclid.org/euclid.em/1062621079