Asymptotic formulas for partial sums of class numbers of indefinite binary quadratic forms

TL;DR

This paper derives asymptotic formulas for partial sums of class numbers of indefinite binary quadratic forms using prime geodesic theorems for congruence subgroups, extending previous results for the modular group.

Contribution

It introduces new asymptotic formulas for partial sums of class numbers by applying prime geodesic theorems to congruence subgroups, broadening the scope beyond the modular group.

Findings

Derived asymptotic formulas for partial sums of class numbers.

Extended previous results to congruence subgroups.

Enhanced understanding of class number distributions in quadratic forms.

Abstract

Sarnak obtained the asymptotic formula of the sum of the class numbers of indefinite binary quadratic forms from the prime geodesic theorem for the modular group. In the present paper, we show several asymptotic formulas of partial sums of the class numbers by using the prime geodesic theorems for the congruence subgroups of the modular group.