Experimental Mathematics

An investigation of bounds for the regulator of quadratic fields

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


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

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

First available in Project Euclid: 3 September 2003

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

Export citation