Subconvexity of Shintani's zeta function

TL;DR

This paper develops a method to prove subconvex bounds for zeta functions counting integral orbits, demonstrated through an estimate for the Shintani zeta function related to class numbers of binary cubic forms.

Contribution

The paper introduces a novel approach to establish subconvexity bounds for zeta functions associated with integral orbit enumeration, specifically applied to the Shintani zeta function.

Findings

Established a subconvexity estimate for the Shintani zeta function

Demonstrated the method on class numbers of binary cubic forms

Contributed to the understanding of orbit counting in arithmetic statistics

Abstract

Enumerating integral orbits in prehomogeneous vector spaces plays an important role in arithmetic statistics. We describe a method of proving subconvexity of the zeta function enumerating the integral orbits, illustrated by proving a subconvex estimate for the Shintani $\zeta$ function enumerating class numbers of binary cubic forms.