Exponential sums formed with the von Mangoldt function and Fourier coefficients of $${ GL}(m)$$ G L ( m ) automorphic forms

Let $$\mu (n)$$ be the Mobius function, $$\Lambda (n)$$ be the von Mangoldt function, and $$A_F(n, 1,\ldots ,1)$$ denote the nth coefficient of the Dirichlet series for L(s, F) associated to a Hecke–Maass form F for $$SL(m,\mathbb {Z})$$ . In this paper, as an appendix to our previous work, we firstly proved that for a Maass form for $$SL(3,\mathbb {Z})$$ , uniformly for all real numbers $$\theta $$ , the sequences $$\{\mu (n)\}$$ and $$\{A_F(n,1)e(n\theta )\}$$ are strong asymptotically orthogonal. Then as an analogue of Baker and Harman’s result on exponential sums formed with the Mobius function under the Generalized Riemann Hypothesis, we investigated the best possible estimates for the sum $$\sum _{n\le x}\Lambda (n) A_F(n, 1,\ldots ,1) e(n^k\theta )$$ under certain assumptions.