Abstract
We study the exponential sums involving Fourier coefficients of Maass forms and exponential functions of the form e(αn β), where 0 ≠ α ∈ ℝ and 0 < β < 1. An asymptotic formula is proved for the nonlinear exponential sum Σ X<n⩽2X λ g (n)e(αn β), when β = 1/2 and |α| is close to 2√q, q ∈ ℤ+, where λ g (n) is the normalized n-th Fourier coefficient of a Maass cusp form for SL 2(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.
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Sun, Q., Wu, Y. Exponential sums involving Maass forms. Front. Math. China 9, 1349–1366 (2014). https://doi.org/10.1007/s11464-014-0360-z
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DOI: https://doi.org/10.1007/s11464-014-0360-z