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An investigation of bounds for the regulator of quadratic fields. (English) Zbl 0859.11057
The authors produced a valuable exposition and status report on the value of the regulator $$R=\log |\varepsilon |$$ for $$\varepsilon (>1)$$ a fundamental unit for a quadratic field of discriminant $$\Delta(>0)$$.
An easy small value of $$R$$ would be $$\log \Delta$$ for $$\Delta= r^2+4$$; but to find large values of $$R$$ (relative to $$\Delta)$$ it is necessary to use formulas such as $$2Rh= \sqrt \Delta L$$, where $$h$$ is the class number and $$L$$ is the Dirichlet $$L$$-function. The $$L$$-function is approximated by partial products of its definition $$\prod (1- (\Delta/p)/p))$$, particularly to find unusually large or small values. Also the $$h$$ is estimated probabilistically by the heuristics of the reviewer and H. W. Lenstra [Lect. Notes Math. 1052, 26-36 (1984; Zbl 0532.12008) and R. A. Mollin and H. C. Williams, Util. Math. 41, 259-308 (1992; Zbl 0757.11036)].
While it is difficult to briefly summarize the many tables and other details, the net effect is that it seems likely that infinitudes of large values of $$\log R\approx \log \sqrt\Delta$$ can be specified.
Reviewer: H.Cohn (Bowie)

##### MSC:
 11R27 Units and factorization 11R11 Quadratic extensions 11Y40 Algebraic number theory computations 11-02 Research exposition (monographs, survey articles) pertaining to number theory
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