The Functional Equations of Langlands Eisenstein Series for $SL(n,\mathbb Z)$

TL;DR

This paper provides an explicit description of Langlands Eisenstein series for SL(n,Z), derives their functional equations from divisor sums and Whittaker functions, and proves a uniqueness conjecture in specific cases.

Contribution

It offers a simple explicit formulation of these Eisenstein series and establishes the uniqueness of their functional equations in certain scenarios.

Findings

Explicit description of Langlands Eisenstein series for SL(n,Z)

Derivation of functional equations from divisor sums and Whittaker functions

Proof of the uniqueness conjecture in specific cases

Abstract

This paper presents a very simple explicit description of Langlands Eisenstein series for ${\rm SL}(n,\mathbb Z)$. The functional equations of these Eisenstein series are heuristically derived from the functional equations of certain divisor sums and certain Whittaker functions that appear in the Fourier coefficients of the Eisenstein series. We conjecture that the functional equations are unique up to a real affine transformation of the $s$ variables defining the Eisenstein series and prove the uniqueness conjecture in certain cases.