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Zapiski Nauchnykh Seminarov POMI, 1993, Volume 204, Pages 11–36
(Mi znsl5781)
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This article is cited in 4 scientific papers (total in 4 papers)
The class numbers of real quadratic fields of discriminant $4p$
E. P. Golubeva
Abstract:
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.
Citation:
E. P. Golubeva, “The class numbers of real quadratic fields of discriminant $4p$”, Analytical theory of numbers and theory of functions. Part 11, Zap. Nauchn. Sem. POMI, 204, Nauka, St. Petersburg, 1993, 11–36; J. Math. Sci., 79:5 (1996), 1277–1292
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https://www.mathnet.ru/eng/znsl5781 https://www.mathnet.ru/eng/znsl/v204/p11
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