On the length of the period of a~quadratic irrationality

E. P. Golubeva

Sbornik. Mathematics, Tome 51 (1985) no. 1, pp. 119-128

Résumé

The estimate $$ Q(D)\asymp\sqrt DL_{4D}(1),\qquad D\to\infty, $$ is proved, where $Q(D)$ is the number of reduced binary quadratic forms of discriminant $4D$ and $L_{4D}(s)=\sum^\infty_{n=1}\bigl(\frac{4D}n\bigr)n^{-s}$ is a Dirichlet $L$-series. Results concerning individual estimates of $l/\log\varepsilon$ are also obtained, where $l$ is the length of the period of the continued-fraction expansion of $\xi\in\mathbf Q(\sqrt D)$ and $\varepsilon$ is a fundamental unit of the field $\mathbf Q(\sqrt D)$. Bibliography: 12 titles.