The distribution of class numbers in a special family of real quadratic fields
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- by Alexander Dahl and Youness Lamzouri PDF
- Trans. Amer. Math. Soc. 370 (2018), 6331-6356 Request permission
Abstract:
We investigate the distribution of class numbers in the family of real quadratic fields $\mathbb {Q}(\sqrt {d})$ corresponding to fundamental discriminants of the form $d=4m^2+1$, which we refer to as Chowla’s family. Our results show a strong similarity between the distribution of class numbers in this family and that of class numbers of imaginary quadratic fields. As an application of our results, we prove that the average order of the number of quadratic fields in Chowla’s family with class number $h$ is $(\log h)/2G$, where $G$ is Catalan’s constant. With minor modifications, one can obtain similar results for Yokoi’s family of real quadratic fields $\mathbb {Q}(\sqrt {d})$, which correspond to fundamental discriminants of the form $d=m^2+4$.References
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Additional Information
- Alexander Dahl
- Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, M3J1P3 Canada
- Email: aodahl@mathstat.yorku.ca
- Youness Lamzouri
- Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, M3J1P3 Canada
- MR Author ID: 804642
- Email: lamzouri@mathstat.yorku.ca
- Received by editor(s): April 21, 2016
- Received by editor(s) in revised form: November 13, 2016
- Published electronically: February 26, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 6331-6356
- MSC (2010): Primary 11R11, 11M20
- DOI: https://doi.org/10.1090/tran/7137
- MathSciNet review: 3814332