Linnik-type problems for automorphic L-functions

AbstractLet m⩾2 be an integer, and π an irreducible unitary cuspidal representation for GLm(AQ), whose attached automorphic L-function is denoted by L(s,π). Let {λπ(n)}n=1∞ be the sequence of coefficients in the Dirichlet series expression of L(s,π) in the half-plane Rs>1. It is proved in this paper that, if π is such that the sequence {λπ(n)}n=1∞ is real, then there are infinitely many sign changes in the sequence {λπ(n)}n=1∞, and the first sign change occurs at some n≪Qπm/2+ε, where Qπ is the conductor of π, and the implied constant depends only on m and ε. This generalizes the previous results for GL2. A result of the same quality is also established for {Λ(n)aπ(n)}n=1∞, the sequence of coefficients in the Dirichlet series expression of −L′L(s,π) in the half-plane Rs>1