The prime number theorem for automorphic L-functions for GLm

Let π be irreducible unitary cuspidal representation of GLm(AQ) with m⩾2, and L(s,π) the L-function attached to π. The prime counting function ψ(x,π) is studied under the Generalized Riemann Hypothesis for L(s,π). It is proved that ψ(x,π)≪x1/22(loglogx), except on a set of x of finite logarithmic measure. Furthermore, the integral mean square of ψ(x,π) is evaluated.