Exponential sums involving automorphic forms for GL(3) over arithmetic progressions

Abstract

Let f be a Hecke-Maass cusp form for SL(3,ℤ) with Fourier coeffcients Af (m, n), and let ϕ(x) be a C∞-function supported on [1, 2] with derivatives bounded by ϕ(j)(x)≪j 1: We prove an asymptotic formula for the nonlinear $$\Sigma_{n\equiv l \rm{mod} \it{q}}$$ Af (m, n)ϕ(n/X)e(3(kn)1/3/q), where e(z) = e2πiz and k ∈ℤ+.