The Generalized Prime Number Theorem for Automorphic L-Functions

Let π and π′ be automorphic irreducible cuspidal representations of GL _m (ℚ) and GL _m′ (ℚ), respectively, and L(s, π × ) be the Rankin-Selberg L-function attached to π and π′. Without assuming the Generalized Ramanujan Conjecture (GRC), the author gives the generalized prime number theorem for L(s, π × ) when π ≅ π′. The result generalizes the corresponding result of Liu and Ye in 2007.