Relations among Dirichlet series whose coefficients are class numbers of binary cubic forms II

TL;DR

This paper explores explicit relations between Dirichlet series linked to class numbers of binary cubic forms, extending previous work and providing new functional equations in self-dual forms.

Contribution

It establishes explicit relations between Dirichlet series of binary cubic forms and their duals, beyond the standard functional equations, for any integral model.

Findings

Dirichlet series satisfy explicit relations to dual series

Functional equations can be expressed in self-dual forms

Results apply to any integral model of binary cubic forms

Abstract

As a continuation of the authors and Wakatsuki's previous paper [5], we study relations among Dirichlet series whose coefficients are class numbers of binary cubic forms. We show that for any integral models of the space of binary cubic forms, the associated Dirichlet series satisfies a simple explicit relation to that of the dual other than the usual functional equation. As an application, we write the functional equations of these Dirichlet series in self dual forms.