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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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References:
[1] Bach E., ”Improved approximations for Euler products” (1994) · Zbl 0824.13009
[2] Chowla S., Proc. London Math. Soc. 50 pp 423– (1949) · Zbl 0032.11006 · doi:10.1112/plms/s2-50.6.423
[3] Cohen H., Séminaire de Théorie des Nombres de Paris 1990–1991 pp 35– (1993)
[4] Cohen H., Number Theory, CUNY, 1982 pp 26– (1983)
[5] Cohen H., Number Theory, Noordwijkerhout, 1983 pp 33– (1984) · doi:10.1007/BFb0099440
[6] Elliot P. D. T. A., J. reine angew. Math. 236 pp 26– (1969)
[7] Halter-Koch F., Abh. Math. Sem. Univ. Hamburg 59 pp 171– (1989) · Zbl 0718.11054 · doi:10.1007/BF02942327
[8] Hooley C., J. reine angew. Math. 353 pp 98– (1984)
[9] Hua L. K., Introduction to Number Theory (1982)
[10] Joshi P. T., J. Number Theory 2 pp 58– (1970) · Zbl 0208.31103 · doi:10.1016/0022-314X(70)90006-5
[11] Landau E., J. London Math. Soc. 11 pp 242– (1936) · Zbl 0015.20001 · doi:10.1112/jlms/s1-11.4.242
[12] Lehmer D. H., Math. Comp. 24 pp 433– (1970)
[13] Lenstra H. W., Number Theory Days pp 123– (1982)
[14] Littlewood J. E., Proc. London Math. Soc. 27 pp 358– (1928) · JFM 54.0206.02 · doi:10.1112/plms/s2-27.1.358
[15] Lukes R. F., Nieuw Archief voor Wiskunde (4) 13 pp 113– (1995)
[16] Lukes R. F., ”Some results on pseudosquares” · Zbl 0852.11072 · doi:10.1090/S0025-5718-96-00678-3
[17] Mollin R. A., Utilitas Math. 41 pp 259– (1992)
[18] Nagell T., Abh. Math. Sem. Univ. Hamburg 1 pp 179– (1922)
[19] Oesterlé J., Journées arithmétiques pp 165– (1978)
[20] Shanks D., Number Theory Institute pp 415– (1969)
[21] Shanks D., Analytic number theory, St. Louis, 1972 pp 267– (1973)
[22] Stephens A. J., Math. Comp. 51 pp 809– (1988) · doi:10.1090/S0025-5718-1988-0958644-7
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