Eisenstein series twisted Shintani zeta function

TL;DR

This paper introduces a twisted zeta function associated with binary cubic forms, proves its meromorphic continuation, and explores its poles and residues, with applications to the distribution of cubic rings.

Contribution

It defines a new twisted zeta function for binary cubic forms and analyzes its analytic properties, extending previous work on zeta functions in number theory.

Findings

Proves meromorphic continuation of the twisted zeta function.

Identifies poles and residues of the zeta function.

Applies results to the equidistribution of cubic rings.

Abstract

We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues. We also identify the poles and residues of the zeta function when restricted to irreducible binary cubic forms. This zeta function can be used to prove the equidistribution of the lattice shape of cubic rings.