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The spectrum of Lévy constants for quadratic irrationalities and class numbers of real quadratic fields. (English. Russian original) Zbl 1130.11333
J. Math. Sci., New York 118, No. 1, 4740-4752 (2003); translation from Zap. Nauchn. Semin. POMI 276, 20-40 (2001).
Summary: Let \(h(d)\) be the class number of the field \(\mathbb{Q}(\sqrt d)\) and let \(\beta(\sqrt d)\) be the Lévy constant. A connection between these constants is studied. It is proved that if \(d\) is large, then the value \(h(d)\) increases, roughly speaking, at the rate \(\beta(\sqrt d)/\beta^2 (\sqrt d)\) as \(\beta(\sqrt d)\) grows. A similar result is obtained in the case where the value \(\beta(\sqrt d)\) is close to \(\log (1+\sqrt 5/2)\), i.e., to the least possible value. In addition, it is shown that the interval \([\log(1+\sqrt 5/2),\log(1+\sqrt 3/\sqrt 2))\) contains no values of \(\beta(\sqrt p)\) for prime \(p\) such that \(p\equiv 3\bmod 4\). As a corollary, a new criterion for the equality \(h(d)=1\) is obtained.

MSC:
11R29 Class numbers, class groups, discriminants
11J70 Continued fractions and generalizations
11K50 Metric theory of continued fractions
11R11 Quadratic extensions
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