Residue of some Eisenstein series

TL;DR

This paper investigates the residue of a specific multivariable Eisenstein series at a critical point, providing explicit formulas for its residue in the context of Siegel modular groups.

Contribution

It offers explicit forms of the residue of the Eisenstein series E_0^{(m)}(z,s) at s=m/2, extending classical results to higher variables.

Findings

Explicit formulas for the residue at s=m/2

Extension of classical Eisenstein series analysis to multiple variables

Insights into the analytic structure of Eisenstein series on Siegel modular groups

Abstract

The real analytic Eisenstein series is a special function that has been studied classically. Its generalization to the case of many variables has been studied extensively. Moreover, the analytic properties of certain Eisenstein series on the Siegel modular groups have also been investigated. The purpose of this study is to provide concrete forms of the residue of E_0^{(m)}(z,s) at s=m/2.