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The average behavior of Fourier coefficients of cusp forms over sparse sequences

Authors: Huixue Lao and Ayyadurai Sankaranarayanan
Journal: Proc. Amer. Math. Soc. 137 (2009), 2557-2565
MSC (2000): Primary 11F30, 11F11, 11F66
Published electronically: March 12, 2009
MathSciNet review: 2497466
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Abstract: Let $ \lambda(n)$ be the $ n$th normalized Fourier coefficient of a holomorphic Hecke eigenform $ f(z)\in S_{k}(\Gamma)$. In this paper we are interested in the average behavior of $ \lambda^2(n)$ over sparse sequences. By using the properties of symmetric power $ L$-functions and their Rankin-Selberg $ L$-functions, we are able to establish that for any $ \varepsilon>0$,

$\displaystyle \sum_{n \leq x}\lambda^2(n^j)=c_{j-1} x+O\left(x^{1-\frac{2}{(j+1)^2+2}+\varepsilon}\right),$

where $ j=2,3,4.$

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Additional Information

Huixue Lao
Affiliation: Department of Mathematics, Shandong Normal University, Jinan Shandong, 250014, People’s Republic of China

Ayyadurai Sankaranarayanan
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400 005, India – and – Institute for Number Theory and Probability Theory, University of Ulm, D-89069, Ulm, Germany

Keywords: Fourier coefficients of cusp forms, symmetric power $L$-function, Rankin-Selberg $L$-function
Received by editor(s): October 17, 2008
Published electronically: March 12, 2009
Additional Notes: This work is supported by the National Natural Science Foundation of China (Grant No. 10701048)
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.