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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on the moments of Kloosterman sums


Authors: Ping Xi and Yuan Yi
Journal: Proc. Amer. Math. Soc. 141 (2013), 1233-1240
MSC (2010): Primary 11L05
Published electronically: September 7, 2012
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Abstract: In this note, we deduce an asymptotic formula for even power moments of Kloosterman sums based on the important work of N. M. Katz on Kloosterman sheaves. In a similar manner, we can also obtain an upper bound for odd power moments. Moreover, we shall give an asymptotic formula for odd power moments of absolute Kloosterman sums. Consequently, we find that there are infinitely many $ a\bmod p$ such that $ S(a,1;p)\gtrless 0$ as $ p\rightarrow +\infty .$


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Additional Information

Ping Xi
Affiliation: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
Email: pingxi.cn@gmail.com

Yuan Yi
Affiliation: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
Email: yuanyi@mail.xjtu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11374-7
PII: S 0002-9939(2012)11374-7
Keywords: Kloosterman sum, mean value, sign change
Received by editor(s): January 18, 2011
Received by editor(s) in revised form: May 19, 2011, July 21, 2011, and August 22, 2011
Published electronically: September 7, 2012
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.