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 \(9/10\).


11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11R47 Other analytic theory
11R45 Density theorems
Full Text: DOI Euclid


[1] N.C. Ankeny and S. Chowla, On the divisibility of the class number of quadratic fields, Pacific J. Math., 5 (1955), 321–324. · Zbl 0065.02402 · doi:10.2140/pjm.1955.5.321
[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
[3] H. Cohen and H.W. Lenstra, Jr, Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), 33–62, (Lecture Notes in Math., 1068, Springer, Berlin, 1984). · Zbl 0558.12002
[4] H. Davenport, Indefinite quadratic forms in many variables. II, Proc. London Math. Soc. (3), 8 (1958), 109–126. · Zbl 0078.03901 · doi:10.1112/plms/s3-8.1.109
[5] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A, 322 (1971), 405–420. · Zbl 0212.08101 · doi:10.1098/rspa.1971.0075
[6] A. Scholz, Über die Beziehung der Klassenzahlen quadratischer Körper zueinander, J. reine angew. Math., 166 (1932), 201–203. · Zbl 0004.05104 · doi:10.1515/crll.1932.166.201
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.