On spectrum Lévy constants for quadratic irrationalities and class numbers of real quadratic fields

E. P. Golubeva

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 17, Tome 276 (2001), pp. 20-40

Résumé

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 $\exp\beta(\sqrt d)/\beta^2(\sqrt d)$ as $\beta(\sqrt d)$ grows. A similar result is obtained in the case where the value $\beta(\sqrt d)$ is close to $\log(1+\sqrt5)/2)$, i.e., to the least possible value. In addition, it is shown that the interval $[\log(1+\sqrt5)/2),\log(1+\sqrt3)/\sqrt2))$ contains no values of $\beta(\sqrt p)$ for prime $p$ such that $p\equiv3\mod4$. As a corollary, a new criterion for the equality $h(d)=1$ is obtained.