The prime number theorem for automorphic L-functions for GLm Original Research Article

Let π be irreducible unitary cuspidal representation of image with mgreater-or-equal, slanted2, 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)much less-thanx1/2(loglogx)2, except on a set of x of finite logarithmic measure. Furthermore, the integral mean square of ψ(x,π) is evaluated.