Exponential sums involving Maass forms

We study the exponential sums involving Fourier coefficients of Maass forms and exponential functions of the form e(αnβ), where 0 ≠ α ∈ ℝ and 0 < β < 1. An asymptotic formula is proved for the nonlinear exponential sum ΣX<n⩽2Xλg(n)e(αnβ), when β = 1/2 and |α| is close to 2√q, q ∈ ℤ+, where λg(n) is the normalized n-th Fourier coefficient of a Maass cusp form for SL2(ℤ). The similar natures of the divisor function τ (n) and the representation function r(n) in the circle problem in nonlinear exponential sums of the above type are also studied.