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On the period length of the continued fraction expansion of quadratic irrationalities and class numbers of real quadratic fields. II. (Russian) Zbl 0693.10019

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.

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
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