Séries d'Eisenstein, intégrales toroïdales et une formule de Hecke. (Eisenstein series, toroidal integrals and a formula of Hecke)
Hecke's integral formula [found in \textit{E.Hecke}'s collected works, 3rd ed. (1983; Zbl 0526.01034), pp. 198-207] writes the Dedekind zeta function as a finite sum over ideal classes of integrals of Epstein zeta- functions over a fundamental domain for units in the number field. It can be viewed as the calculation of the zeroth Fourier coefficient of the Epstein zeta-function, and this led Siegel to generalize the result to obtain analogous formulas for Hecke L-functions with grössencharacters [see \textit{C. L. Siegel}'s ''Lectures on advanced analytic number theory'' (1965; Zbl 0278.10001)]. An exposition of Hecke's formula can be found in the reviewer's ''Harmonic analysis on symmetric spaces and applications'', Vol. I (1985; Zbl 0574.10029), pp. 70-73. The paper under review gives a generalized adelic version of the integral formula. Another adelic formulation of Hecke's integral formula can be found in \textit{L. J. Goldstein}'s paper [Isr. J. Math. 17, 283-301 (1974; Zbl 0292.12013)].
scientific article; zbMATH DE number 3929112