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The distribution of class numbers in a special family of real quadratic fields


Authors: Alexander Dahl and Youness Lamzouri
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11R11, 11M20
DOI: https://doi.org/10.1090/tran/7137
Published electronically: February 26, 2018
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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$.


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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
Email: lamzouri@mathstat.yorku.ca

DOI: https://doi.org/10.1090/tran/7137
Received by editor(s): April 21, 2016
Received by editor(s) in revised form: November 13, 2016
Published electronically: February 26, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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