Mathematical analysis reveals limiting behaviors of modular function cycle integrals at real quadratic irrationalities, indicating new class number distributions for specific quadratic fields.
In this paper, we study the values of modular functions at real quadratic irrationalities which are defined in terms of cycle integrals of modular functions associated with closed geodesics. We refine a result of Bengoechea and Imamog̎lu to include values (and its limiting behaviour) of modular functions at a larger class of real quadratic irrationalities. As a consequence, we deduce the limiting behaviour of values of modular functions at powers of Pisot–Vijayaraghavan numbers. Further, we deduce distribution of class numbers of real quadratic fields having discriminants of a particular type.
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Kumar et al. (2026) studied this question.
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