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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The average behavior of Fourier coefficients of cusp forms over sparse sequences
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by Huixue Lao and Ayyadurai Sankaranarayanan PDF
Proc. Amer. Math. Soc. 137 (2009), 2557-2565 Request permission

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
  • 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
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2497466