First moments of Fourier coefficients of GL(r) cusp forms

Abstract

Let π be an irreducible cuspidal automorphic representation of GL(r,A) where r≥2 and A is the adele ring of Q. Let aπ(n) denote the nth Dirichlet series coefficient of the L-function associated to π. The main goal of this paper is to obtain strong bounds for the first moment ∑n≤xaπ(n) as x→∞. The bounds we obtain are better than all previously obtained bounds for the higher rank situation when r≥3 and π is not a symmetric power of a GL(2,A) cuspidal automorphic representation.