The first coefficient of Langlands Eisenstein series for ${\rm SL}(n,\mathbb Z)$
Volume 214 / 2024
Acta Arithmetica 214 (2024), 179-189
MSC: Primary 11F55; Secondary 11F72
DOI: 10.4064/aa230412-15-7
Published online: 21 December 2023
Abstract
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.
