A generalization of Kronecker's first limit formula to GL(n)

TL;DR

This paper extends Kronecker's first limit formula from SL(2,Z) to higher dimensions, providing new insights into Eisenstein series and Dedekind zeta functions for number fields.

Contribution

It generalizes Kronecker's limit formula to maximal parabolic Eisenstein series for SL(n,Z), linking it to Dedekind zeta functions of number fields.

Findings

Derived the polar and constant terms of Eisenstein series for SL(n,Z)

Connected the generalized formula to Dedekind zeta functions at s=1

Extended classical results to higher-dimensional cases

Abstract

Kronecker's first limit formula gives the polar and constant terms of the Laurent series expansion of the Eisenstein series for SL(2,Z) at s=1. In this article, we generalize the formula to certain maximal parabolic Eisenstein series associated to SL(n,Z) for n greater than or equal to 2. We also show how the generalized formula can be used to give the polar and constant terms of the Dedekind zeta function of any number field at s=1.