Exponential sums involving Maass forms

Abstract

We study the exponential sums involving Fourier coefficients of Maass forms and exponential functions of the form e(αn^β),where \mathrm{0}\ne \alpha \in ℝ and 0<β<1. An asymptotic formula is proved for the nonlinear exponential sum , when β = 1/2 and |α| is close to \mathrm{2}\sqrt{q}, q\in {\mathbb{Z}}^{+}, where λ_g(n) is the normalized n-th Fourier coefficient of a Maass cusp form for SL₂(\mathbb{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.

Keywords

Fourier coefficients of Maass form / nonlinear exponential sum / number-theoretic function

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Higher Education Press and Springer-Verlag Berlin Heidelberg