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

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.