Asymptotic formulas for class number sums of indefinite binary quadratic forms in arithmetic progressions

TL;DR

This paper derives simpler and sharper asymptotic formulas for class number sums of indefinite binary quadratic forms in arithmetic progressions, extending previous results with improved methods and expressions.

Contribution

It introduces new asymptotic formulas for class number sums in arithmetic progressions using a Tchebotarev-type prime geodesic theorem, with simpler leading terms and sharper error estimates.

Findings

Derived asymptotic formulas for class number sums in arithmetic progressions.

Provided simpler expressions for leading terms compared to previous work.

Achieved sharper estimates of the remainder terms.

Abstract

It is known that there is a one-to-one correspondence between equivalence classes of primitive indefinite binary quadratic forms and primitive hyperbolic conjugacy classes of the modular group. Due to such a correspondence, Sarnak obtained the asymptotic formula for the class number sum in order of the fundamental unit by using the prime geodesic theorem for the modular group. In the present paper, we propose asymptotic formulas of the class number sums over discriminants in arithmetic progressions. Since there are relations between the arithmetic properties of the discriminants and the conjugacy classes in the finite groups given by the modular group and its congruence subgroups, we can get the desired asymptotic formulas by arranging the Tchebotarev-type prime geodesic theorem. While such asymptotic formulas were already given by Raulf, the approaches are quite different, the expressions of the leading terms of our asymptotic formulas are simpler and the estimates of the reminder terms are sharper.