Improving a result of Montgomery and Weinberger, we establish
the existence of infinitely many real quadratic fields for which the
class numbers are as large as possible. These values are achieved using
a special family of fields, first studied by Chowla. In a subsequent
work, joint with A. Dahl, we investigate the distribution of class
numbers in Chowla’s family, and show a strong similarity between this
distribution and that of class numbers of imaginary quadratic fields,
previously studied by Granville and Soundararajan. As an application of
these methods, we determine the average order of the number of quadratic
fields with class number $h$ in several families including Chowla's
family of real quadratic fields, and the family of imaginary quadratic
fields with prime discriminants.
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