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Proceedings of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9939-09-09855-4
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 $\varepsilon >0$, \[ \sum _{n \leq x}\lambda ^2(n^j)=c_{j-1} x+O\left (x^{1-\frac {2}{(j+1)^2+2}+\varepsilon }\right ),\] 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

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.