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Class numbers of indefinite binary quadratic forms and the residual indices of integers modulo \(p\). (English. Russian original) Zbl 1077.11068

J. Math. Sci., New York 122, No. 6, 3685-3698 (2004); translation from Zap. Nauchn. Semin. POMI 286, 179-199 (2002).
Summary: Let \(h(d)\) be the class number of properly equivalent primitive binary quadratic forms \(ax^2+ bxy+ cy^2\) with discriminant \(d=b^2-4ac\). The behavior of \(h(5p^2)\), where \(p\) runs over primes, is studied. It is easy to show that there are few discriminants of the form \(5p^2\) with large class numbers. In fact, one has the estimate \(\#\{p\leq x\mid h(5p^2)> x^{1-\delta}\}\ll x^{2\delta}\), where \(\delta\) is an arbitrary constant number in \((0;1/2)\).
Assume that \(\alpha(x)\) is a positive function monotonically increasing for \(x\to\infty\) and \(\alpha(x)\to\infty\). If \(\alpha(x)\leq (\log x)(\log\log x)^{-3}\), then (assuming the validity of the extended Riemann hypothesis for certain Dedekind zeta-functions) it is proved that \[ \#\left\{ p\leq x\,\biggl|\, \biggl( \frac5p\biggr)= 1,\;h(5p^2)> \alpha(x) \right\} \asymp \frac{\pi(x)}{\alpha(x)}. \] It is also proved that for an infinite set of \(p\) with \({\frac 5p}=1\) one has the inequality \(h(5p^2)\geq \frac {\log\log p}{\log_k p}\), where \(\log_k p\) is the \(k\)-fold iterated logarithm (\(k\) is an arbitrary integer, \(k\geq 3\)). Results on mean values of \(h(5p^2)\) are also obtained. Similar facts are true for the residual indices of an integer \(a\geq 2\) modulo \(p\): \(r(a,p)= \frac{p-1}{o(a,p)}\), where \(o(a,p)\) is the order of \(a\) modulo \(p\).

MSC:

11N37 Asymptotic results on arithmetic functions
11E41 Class numbers of quadratic and Hermitian forms
11R45 Density theorems
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