Frontiers of Mathematics in China >

2014 , Vol. 9 >Issue 6: 1349 - 1366

DOI: https://doi.org/10.1007/s11464-014-0360-z

RESEARCH ARTICLE

Exponential sums involving Maass forms

  • Qingfeng SUN 1 ,
  • Yuanying WU , 1
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  • 1. School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, China
  • 2. School of Mathematics, Shandong University, Jinan 250100, China

Received date: 12 Dec 2013

Accepted date: 14 Jan 2014

Published date: 29 Oct 2014

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
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Abstract

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 ∑X<n≤2Xλg(n)e(αnβ), when β = 1/2 and |α| is close to 2q, 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.

Key words： Fourier coefficients of Maass form; nonlinear exponential sum; number-theoretic function

Cite this article

Qingfeng SUN , Yuanying WU . Exponential sums involving Maass forms[J]. Frontiers of Mathematics in China, 2014 , 9(6) : 1349 -1366 . DOI: 10.1007/s11464-014-0360-z

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