The Spectrum of Lévy Constants for Quadratic Irrationalities and Class Numbers of Real Quadratic Fields

Abstract

Let h(d) be the class number of the field \(\mathbb{Q}(\sqrt d )\) and let \(\beta (\sqrt d )\) be the Levy 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 \(\left[ {\log (1 + \sqrt 5 /2),\log (1 + \sqrt 3 /\sqrt 2 ))} \right.\) contains no values of \(\beta (\sqrt p )\) for prime p such that p ≡3 (mod 4). As a corollary, a new criterion for the equality h(d)=1 is obtained. Bibliography: 14 titles.