Golubeva, E. P. On the period length of the continued fraction expansion of quadratic irrationalities and class numbers of real quadratic fields. II. (Russian) Zbl 0693.10019 Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 168, 11-22 (1988). Let \(h(m)\) be the class number of a binary quadratic form with determinant \(m\); \(d\) be a fixed integer, \(d\ne m^2\), and \(p\) be a prime. The paper proves that \[ \sum_{p\leq x}h(dp^2) = O_d(x^{3/2}). \] This is a direct generalization of the author’s earlier result [ibid. 160, 72–81 (1987; Zbl 0633.10022)]. By a slight modification of considerations by R. Gupta and M. Ram Murty [Invent. Math. 78, 127–130 (1984; Zbl 0549.10037)] and D. R. Heath-Brown [Q. J. Math., Oxf. II. Ser. 37, 27–38 (1986; Zbl 0586.10025)] the author proves that among the primes \(p\le x\) there exist not less than \(Cx \log^2x\) ones for which at least one from \(h(7p^2)\), \(h(11p^2)\), \(h(19p^2)\) equals 2. The paper also considers fundamental discriminants and gives a lower bound for a fundamental unit \(\varepsilon(p)\) of \(\mathbb{Q}(\sqrt{p})\) \[ \varepsilon(p) \gg p^2 \log^{-c}p \] for all \(p\equiv 3\pmod 4\), \(p\le x\) with \(O(x \log^{-2}x)\) possible exceptions. The proof is based on the investigation of the period length of the continued fraction expansion of quadratic irrationalities. At the end of the paper F. Hirzebruch’s formula [Enseign. Math., II. Ser. 19, 183–281 (1973; Zbl 0285.14007)] connecting \(h(-p)\), \(p\equiv 3\pmod 4\) with the continued fraction expansion of \(\sqrt{p}\) is generalized to the case of non-prime discriminants. Reviewer: Guram Gogishvili (Tbilisi) Cited in 1 ReviewCited in 3 Documents MSC: 11E41 Class numbers of quadratic and Hermitian forms 11A55 Continued fractions 11R27 Units and factorization 11R29 Class numbers, class groups, discriminants 11R11 Quadratic extensions 11E16 General binary quadratic forms Keywords:class number; binary quadratic form; fundamental discriminants; lower bound; fundamental unit; period length; continued fraction expansion; quadratic irrationalities Citations:Zbl 0633.10022; Zbl 0549.10037; Zbl 0586.10025; Zbl 0285.14007 PDFBibTeX XMLCite \textit{E. P. Golubeva}, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 168, 11--22 (1988; Zbl 0693.10019) Full Text: EuDML