Linnik-type problems for automorphic L-functions

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