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

 

Mean value of mixed exponential sums


Author: Huaning Liu
Journal: Proc. Amer. Math. Soc. 136 (2008), 1193-1203
MSC (2000): Primary 11L03, 11L05
Published electronically: December 18, 2007
MathSciNet review: 2367093
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Abstract: For integers $ q$, $ m$, $ n$, $ k$ with $ q,k\geq 1$, and Dirichlet character $ \chi\bmod q$, we define a mixed exponential sum

$\displaystyle C(m,n,k,\chi;q):= \mathop{{\sum}'}_{a=1}^q\chi(a)\hbox{e}\left(\frac{ma^k+na}{q}\right), $

where $ \displaystyle \hbox{e}(y)=\hbox{e}^{2\pi iy}$, and $ \sum'_{a}$ denotes the summation over all $ a$ with $ (a,q)=1$. The main purpose of this paper is to study the mean value of

$\displaystyle \sum_{\chi\bmod q}\mathop{{\sum}'}_{m=1}^q\left\vert C(m,n,k,\chi;q)\right\vert^4, $

and to give a related identity on the mean value of the general Kloosterman sum

$\displaystyle K(m,n,\chi;q):=\mathop{{\sum}'}_{a=1}^q\chi(a)\hbox{e}\left(\frac{ma +n\overline{a}}{q}\right), $

where $ a\overline{a} \equiv 1 \bmod q$.


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Additional Information

Huaning Liu
Affiliation: Department of Mathematics, Northwest University, Xi’an, Shaanxi, People’s Republic of China
Email: hnliu@nwu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09075-2
PII: S 0002-9939(07)09075-2
Keywords: Exponential sum, Kloosterman sum, mean value, identity.
Received by editor(s): July 26, 2006
Received by editor(s) in revised form: January 9, 2007
Published electronically: December 18, 2007
Additional Notes: This work was supported by the National Natural Science Foundation of China under Grant No.60472068 and No.10671155; Natural Science Foundation of Shaanxi province of China under Grant No.2006A04; and the Natural Science Foundation of the Education Department of Shaanxi Province of China under Grant No.06JK168.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.