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Quadratic class numbers divisible by 3. (English) Zbl 1140.11050

It follows from a result of K. Soundararajan [J. Lond. Math. Soc. (2) 61, No. 3, 681–690 (2000; Zbl 1018.11054)] that the number of square-free numbers \(0<d\leq x\) for which the class-number of the field \(\mathbb Q(\sqrt{-d})\) is divisible by \(3\) is for every \(\varepsilon>0\) greater than \(B(\varepsilon)x^{7/8-\varepsilon}\) (with positive \(B(\varepsilon)\)), and his method has been utilized by D. Byeon and E. Koh [Manuscr. Math. 111, 261–263 (2003; Zbl 1125.11060)] to get the same assertion for real quadratic fields. In the reviewed paper an improvement of Soundararajan’s approach is presented, which in both cases leads to the replacement of \(7/8\) by \(9/10\).

MSC:

11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11R47 Other analytic theory
11R45 Density theorems
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References:

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[2] D. Byeon and E. Koh, Real quadratic fields with class number divisible by 3, Manuscripta Math., 111 (2003), 261–263. · Zbl 1125.11060 · doi:10.1007/s00229-003-0379-z
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[7] K. Soundararajan, Divisibility of class numbers of imaginary quadratic fields, J. London Math. Soc., 61 (2000), 681–690. · Zbl 1018.11054 · doi:10.1112/S0024610700008887
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