Orbital L-functions for the space of binary cubic forms

TL;DR

This paper introduces orbital L-functions for binary cubic forms, explores their analytic properties, and uses these results to establish secondary terms in counting functions for cubic fields.

Contribution

It defines orbital L-functions in the context of binary cubic forms and analyzes their functional equations and residues, advancing understanding of their analytic structure.

Findings

Orbital L-functions exhibit specific functional equations.

Residue formulas for orbital L-functions are derived.

Results support the existence of secondary terms in cubic field counting functions.

Abstract

We introduce the notion of orbital L-functions for the space of binary cubic forms and investigate their analytic properties. We study their functional equations and residue formulas in some detail. Aside from the intrinsic interest, results from this paper are used to prove the existence of secondary terms in counting functions for cubic fields. This is worked out in a companion paper (arXiv:1102.2914).