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).