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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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