Lengths of the periods of the continued fraction expansion of quadratic irrationalities and on the class numbers of real quadratic fields

It is proved that the relation h(d)=2 is valid for at least Cx^1/2 log^–2 x values of dx. 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 p3 (mod 4), px, for (p), a fundamental unit of field and l(p), the length of the period of the continued fraction expansion of p, we have estimates (p)p² log^–c p, l(p)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 3 (mod 4), with the continued fraction expansion of _p (Astérisque, no. 24–25, pp. 81–97, 1975; MR 51 #10293).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 168, pp. 11–22, 1988.