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

摘要

Let f be a Hecke-Maass cusp form for SL(3,Z) with Fourier coefficients 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 exponential sum Σn≡l mod qAf(m,n)φ(n/X)e(3(kn)1/3/q),where e(z) =e2πiz and k ∈ Z+.