The class numbers of real quadratic fields of discriminant $4p$

E. P. Golubeva

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 11, Tome 204 (1993), pp. 11-36

Résumé

For $p$ prime, $p=3\,(\operatorname{mod}4)$, we study the expansion of $\sqrt p$ into a continued fraction. In particular, we show that in the expansion $$ \sqrt p=[n,\overline{l_1,\dots,l_L,l,l_L,\dots,l_1,2n}] $$ $l_1,\dots,l_L$ satisfy at least $L/2$ linear relations. We also obtain a new lower bound for the fundamental unit $\varepsilon_p$ of the field $\mathbb Q(\sqrt p)$ for almost all $p$ under consideration: $\varepsilon_p>p^3/\log^{1+\delta}p$ for all $p\ge x$ with $O(x/\log^{1+\delta}x)$ possible exceptions (here $\delta>0$ is an arbitrary constant), and an estimate for the mean value of the class number of $\mathbb Q(\sqrt p)$ with respect to averaging over $\varepsilon_p$: $$ \sum_{p\equiv3\,(\operatorname{mod}4),\ \varepsilon_p\le x}h(p)=O(x). $$ Bibliography: 11 titles.