Subconvexity of Shintani’s zeta function

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 ζ \zeta function enumerating class numbers of binary cubic forms.

Subconvexity of twisted Shintani zeta functions

“…In a previous paper [14], the authors proved the subconvexity of Shintani's zeta function enumerating class numbers of binary cubic forms. The purpose of this article is to demonstrate that the same approach proves subconvexity for the Maass cusp form twisted version of the zeta function [12].…”

“…This can be compared to an error term of τ 98 99 +ǫ for the untwisted zeta function, proved in [14].…”