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

Author(s): Huixue Lao; Ayyadurai Sankaranarayanan
Journal: Proc. Amer. Math. Soc. 137 (2009), 2557-2565.
MSC (2000): Primary 11F30, 11F11, 11F66
Posted: March 12, 2009
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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
Email: laohuixue@sina.com

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
Email: sank@math.tifr.res.in, ayyadurai.sankaranarayanan@uni-ulm.de

DOI: 10.1090/S0002-9939-09-09855-4
PII: S 0002-9939(09)09855-4
Keywords: Fourier coefficients of cusp forms, symmetric power $L$-function, Rankin-Selberg $L$-function
Received by editor(s): October 17, 2008
Posted: 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
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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