The Shintani double zeta functions

TL;DR

This paper derives an explicit formula for Shintani double zeta functions over arbitrary number fields, revealing new functional equations, holomorphicity results, and asymptotic formulas related to quadratic Dirichlet L-functions.

Contribution

It provides the most general explicit formula for Shintani double zeta functions with applications to functional equations, holomorphicity, and asymptotic analysis.

Findings

Derived a new functional equation for Shintani double zeta functions.

Proved holomorphicity of a related Dirichlet series in the context of automorphic forms.

Established an asymptotic formula for averages of quadratic Dirichlet L-values.

Abstract

In this paper, we give an explicit formula of the Shintani double zeta functions with any ramification in the most general setting of adeles over an arbitrary number field. Three applications of the explicit formula are given. First, we obtain a functional equation satisfied by the Shintani double zeta functions in addition to Shintani's functional equations. Second, we establish the holomorphicity of a certain Dirichlet series generalizing a result by Ibukiyama and Saito. This Dirichlet series occurs in the study of unipotent contributions of the geometric side of the Arthur-Selberg trace formula of the symplectic group. Third, we prove an asymptotic formula of the weighted average of the central values of quadratic Dirichlet $L$-functions.