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Twisted higher moments of Kloosterman sums


Author: Chunlei Liu
Journal: Proc. Amer. Math. Soc. 130 (2002), 1887-1892
MSC (2000): Primary 11L05
DOI: https://doi.org/10.1090/S0002-9939-02-06510-3
Published electronically: February 8, 2002
MathSciNet review: 1896019
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Abstract: Let $\chi $ be a nontrivial Dirichlet character modulo an odd prime $p$. Write

\begin{displaymath}S(a)=\sum \limits _{x=1}^{p-1}e(\frac{x+ax^{-1}}{p})=2\sqrt {p}\cos \theta (a). \end{displaymath}

We shall prove

\begin{displaymath}\sum \limits _{a=1}^{p-1}\chi (a)S(a)^{2}=\chi (-1)g(\chi )^{2}J(\chi ,\bar {\chi }^{2}) \end{displaymath}

and, for complex $\chi $,

\begin{displaymath}\vert\sum \limits _{a=1}^{p-1}\chi (a)\frac{\sin (k+1)\theta (a)}{\sin \theta (a)}\vert\leq c(k)\sqrt {p}, k>0, \end{displaymath}

where $c(k)$ is a constant depending only on $k$.


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Additional Information

Chunlei Liu
Affiliation: Morningside Center of Mathematics, Chinese Academy of Science, Beijing 100080, People’s Republic of China
Address at time of publication: P. O. Box 1001-745, Zhengzhou 450002, People’s Republic of China
Email: chunleiliu@mail.china.com

DOI: https://doi.org/10.1090/S0002-9939-02-06510-3
Keywords: Kloosterman sum, Dirichlet character
Received by editor(s): September 19, 2000
Published electronically: February 8, 2002
Additional Notes: This research is supported by MCSEC and NSFC
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2002 American Mathematical Society

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