On relations among Dirichlet series whose coefficients are class numbers of binary cubic forms

TL;DR

This paper investigates the class numbers of integral binary cubic forms, classifies invariant lattices, explores relationships between associated Dirichlet series, and analyzes their analytic properties and functional equations.

Contribution

It classifies invariant lattices and establishes explicit relationships between Dirichlet series for binary cubic forms, enhancing understanding of their analytic and functional properties.

Findings

Classification of invariant lattices for binary cubic forms

Explicit relationships between Dirichlet series of different lattices

Reformulation of the functional equation in a self-dual form

Abstract

We study the class numbers of integral binary cubic forms. For each $SL_2(Z)$ invariant lattice $L$, Shintani introduced Dirichlet series whose coefficients are the class numbers of binary cubic forms in $L$. We classify the invariant lattices, and investigate explicit relationships between Dirichlet series associated with those lattices. We also study the analytic properties of the Dirichlet series, and rewrite the functional equation in a self dual form using the explicit relationship.