RESEARCH ARTICLE

Sums of Fourier coefficients of cusp forms of level D twisted by exponential functions

  • Huan LIU
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  • School of Mathematics, Shandong University, Jinan 250100, China

Received date: 15 Dec 2015

Accepted date: 04 Mar 2016

Published date: 20 Apr 2017

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

Let g be a holomorphic or Maass Hecke newform of level D and nebentypus χD, and let ag(n) be its n-th Fourier coefficient. We consider the sum S1=X<n2Xag(n)e(anβ) and prove that S1 has an asymptotic formula when β = 1/2 and αis close to ±2q/D for positive integer qX/4 and X sufficiently large. And when 0<β<1 and α, β fail to meet the above condition, we obtain upper bounds of S1. We also consider the sum S2=n>0ag(n)e(anβ)ϕ(n/X) with ϕ(x)Cc(0,+) and prove that S2 has better upper bounds than S1 at some special α and β.

Cite this article

Huan LIU . Sums of Fourier coefficients of cusp forms of level D twisted by exponential functions[J]. Frontiers of Mathematics in China, 2017 , 12(3) : 655 -673 . DOI: 10.1007/s11464-016-0534-y

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