The first coefficient of Langlands Eisenstein series for ${\rm SL}(n,\mathbb Z)$

Fourier coefficients of Eisenstein series figure prominently in the study of automorphic L-functions via the Langlands–Shahidi method, and in various other aspects of the theory of automorphic forms and representations.

In this paper, we define Langlands Eisenstein series for ${\rm SL}(n,\mathbb Z)$ in an elementary manner, and then determine the first Fourier coefficient of these series in a very explicit form. Our proofs and derivations are short and simple, and use the Borel Eisenstein series as a template to determine the first Fourier coefficient of other Langlands Eisenstein series.