On some estimates involving Fourier coefficients of Maass cusp forms.

Let f be a Hecke–Maass cusp form for SL 2 (ℤ) with Laplace eigenvalue λ f (Δ) = 1 / 4 + μ 2 and let λ f (n) be its n th normalized Fourier coefficient. It is proved that, uniformly in α , β ∈ ℝ , ∑ n ≤ X λ f (n) e (α n 2 + β n) ≪ X 7 / 8 + λ f (Δ) 1 / 2 + , where the implied constant depends only on . We also consider the summation function of λ f (n) and under the Ramanujan conjecture we are able to prove ∑ n ≤ X λ f (n) ≪ X 1 / 3 + λ f (Δ) 4 / 9 + with the implied constant depending only on . [ABSTRACT FROM AUTHOR]