On the period length of the continued fraction expansion of quadratic irrationalities and class numbers of real quadratic fields.~II

E. P. Golubeva

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 9, Tome 168 (1988), pp. 11-22

Résumé

It is proved that the relation $h(d)=2$ is valid for at least $Cx^{1/2}\log^{-2}x$ values of $d\leq x$. Here $h(d)$ is the number of the classes of binary quadratic forms of determinant $d$, while $C>0$ is a constant. Further, it is shown that for almost all primes $p\equiv3\,(\operatorname{mod}4)$, $p\leq x$, for $\varepsilon(p)$, a fundamental unit of field $\mathbb{Q}(\sqrt{p})$ and $\ell(p)$, the length of the period of the continued fraction expansion of $\sqrt{p}$, we have estimates $\varepsilon(p)\gg p^2\log^{-c}p$, $\ell(p)\gg\log p$, which improve a result of Hooley (J. Reine Angew. Math., Vol. 353, pp. 98–131, 1984; MR 86d:11032). In addition, a generalization is given to composite discriminants of the Hirzebruch–Zagier formula, relating $h(-p)$, $p\equiv3\,(\operatorname{mod}4)$, with the continued fraction expansion of $\sqrt{p}$ (Asterisque, no. 24–25, pp. 81–97, 1975; MR 51 10293).