On the period length of the continued fraction expansion of a quadratic irrational and the class number of real quadratic fields

E. P. Golubeva

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 8, Tome 160 (1987), pp. 72-81

Résumé

The fundamental result of the paper is the following theorem: suppose that the Riemann conjecture is valid for the Dedekind $\xi$-functions of all fields $\mathbb{Q}\Bigl(\Bigl(\frac{1+\sqrt{5}}{2}\Bigr)^{1/k},1^{1/k}\Bigr)$ Then there exists a constant $C>0$ such that on the interval $p\leq x$ one can find at least $Cx\log^{-1}x$ prime numbers $p$ for which $h(Sp^2)=2$. Here $h(d)$ is the number of proper equivalence classes of primitive binary quadratic forms of discriminant $d$. In addition, it is proved that $$ \sum_{p\leq x}h(Sp^2)\log p=O(x^{3/2}). $$ For these sequence of discriminants of a special form with increasing square-free part, one has obtained a nontrivial estimate from above for the number of classes.