In this paper we define a Rankin-Selberg L-function attached to automorphic cuspidal representations of GLm(A E ) × GL m (A F ) over cyclic algebraic number fields E and F which are invariant under the Galois action, by exploiting a result proved by Arthur and Clozel, and prove a prime number theorem for this L-function.

In this paper we define a Rankin–Selberg L-function attached to two Galois invariant automorphic cuspidal representations of GL m(𝔸E) and GL m′(𝔸F) over cyclic Galois extensions E and F of prime degree. This differs from the classical case in that the two extension fields E and F could be completely unrelated to one another, and we exploit the existence of the automorphic induction functor over cyclic extensions (see [J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Mathematics Studies, No. 120 (Princeton University Press, Princeton, NJ, 1989)]) to define the L-function. Using a result proved by C. S. Rajan, we prove a prime number theorem for this L-function, and proceed to calculate the n-level correlation function of high nontrivial zeros of a product L(s, π1)L(s, π2)…L(s, πk) where πi is a Galois invariant cuspidal representation of GL ni(𝔸Fi) with Fi a cyclic Galois extension of prime degree ℓi for i = 1,…,k, thus generalizing the results of Liu and Ye [Functoriality of automorphic L-functions through their zeros, Sci. China Ser. A51(1) (2008) 1–16].

Given two unitary automorphic cuspidal representationsπandπ′defined onGLm(𝔸E)andGLm'(𝔸F), respectively, withEandFbeing Galois extensions ofℚ, we consider two generalized Rankin-SelbergL-functions obtained by forcefully factoringL(s,π)  and  L(s,π'). We prove the absolute convergence of theseL-functions forRe(s)>1. The main difficulty in our case is that the two extension fields may be completely unrelated, so we are forced to work either “downstairs” in some intermediate extension betweenE∩Fandℚ, or “upstairs” in some extension field containing the composite extensionEF. We close by investigating some special cases when analytic continuation is possible and show that when the degrees of the extension fieldsEandFare relatively prime, the two different definitions give the same generating function.

In this paper we define a Rankin-Selberg L-function attached to automorphic cuspidal representations of GLm(A E )×GL m ′ (A F ) over solvable algebraic number fields E and F which are invariant under the Galois action, using a result proved by C.S. Rajan, and prove a prime number theorem for this L-function.