Fourier coefficients of Eisenstein series of the exceptional group of type G₂

Let F be a number fields and K be a commutative algebra over F of degree n. A basic question in number theory is whether the ratio ζK(s)/ζF(s) of the two Dedekind zeta functions is an entire function in the complex variable s. From the point of view of the trace formula, the above basic question is expected to be equivalent to a basic question in automorphic L-functions, which asks whether or not the ratio L^S(Π⊗Π^V,s)/ζ^S_F(s) is entire for all irreducible cuspidal automorphic representation of GL(n, double-struck A_F) with trivial central character, where L^S(Π⊗Π^V,s) is the standard tensor product L-function of Π with its contragredient Π^V, see for example the work of Jacquet and Zagier [JaZa]. The main idea in this paper is to develop two intrinsically related methods to attack the above two questions. The work of Siegel [Sie], and of Shimura [Shi] (and of Gelbart and Jacquet [GeJa]) provided an evidence for this approach for the case of n = 2. Combined with the work of Ginzburg [Gin], the main result of this paper shows that our approach works for the case of n = 3. It is hoped that such an approach extends to at least the case of n = 5.