Strong orthogonality between the Mobius function, additive characters and the coefficients of the L-functions of degree three

Let F be a self-dual Hecke-Maass cusp form for SL(3, Z) and let a(F)(1, n) denote the n-th coefficient of the Godement-Jacquet L-function L(s, F). Then we show that there exists an absolute constant c(0) > 0 such that Sigma(n <= X) a(F)(1, n)mu(n)e(n alpha) << X exp (-c(0)root log X). Here the implied constant depends only on the form F and the bound is uniform in alpha is an element of R. Moreover, we notice that the aforementioned result generalises to self-dual automorphic cuspidal representations of GL(3)(A(Q)), with unitary central character. (C) 2019 Elsevier Inc. All rights reserved.