Selberg＇s Normal Density Theorem for Automorphic L-Functions for GLm

Let π be an irreducible unitary cuspidal representation of GL m \({(\Bbb {A}_\Bbb{Q})}\) with m ≥ 2, and L(s, π) the L-function attached to π. Under the Generalized Riemann Hypothesis for L(s, π), we estimate the normal density of primes in short intervals for the automorphic L-function L(s, π). Our result generalizes the corresponding theorem of Selberg for the Riemann zeta-function.