Class Numbers of Indefinite Binary Quadratic Forms and the Residual Indices of Integers Modulo p

ISSN： 1072-3374

Source： Journal of Mathematical Sciences, Vol.122, Iss.6, 2004-08, pp. : 3685-3698

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Abstract

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\le x\,|\,h(5p^2) \gt 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 $\alpha(x)\le(\log x)(\log\log x)^{-3},$ then (assuming the validity of the extended Riemann hypothesis for certain Dedekind zeta-functions) it is proved that \#\bigg\{p\le x\,\Big|\,\bigg(\frac5p\bigg)=1,h(5p^2) \gt \alpha(x)\bigg\} \asymp\frac{\pi(x)}{\alpha(x)}. It is also proved that for an infinite set of p with \big(\f5p\big)=1 one has the inequality $h(5p^2)\ge\frac{\log\log p}{\log_k p},$ where $\log_k p$ is the k-fold iterated logarithm (k is an arbitrary integer, $k\ge3$ ). Results on mean values of h(5p^2) are also obtained. Similar facts are true for the residual indices of an integer $a\ge2$ modulo p: $r(a,p)=\frac{p-1}{o(a,p)},$ where o(a,p) is the order of a modulo p. Bibliography: 13 titles.