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Explicit values of multi-dimensional Kloosterman sums for prime powers, II

Author: S. Gurak
Journal: Math. Comp. 77 (2008), 475-493
MSC (2000): Primary 11L05, 11T24
Published electronically: May 14, 2007
MathSciNet review: 2353962
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Abstract: For any integer $ m>1$ fix $ \zeta_{m}={\rm exp}(2 \pi i/m)$, and let $ {\bf Z}_{m}^{*}$ denote the group of reduced residues modulo $ m$. Let $ q=p^{\alpha}$, a power of a prime $ p$. The hyper-Kloosterman sums of dimension $ n>0$ are defined for $ q$ by

$\displaystyle \displaylines{ R(d,q)= \sum_{x_{1}, ..., x_{n} \in Z_{q}^{*}} \ze... ...ots +x_{n} +d(x_{1} \cdots x_{n})^{-1}} \;\;\;\;\;\; (d \in {\bf Z}_{q}^{*}), }$

where $ x^{-1}$ denotes the multiplicative inverse of $ x$ modulo $ q$.

Salie evaluated $ R(d,q)$ in the classical setting $ n=1$ for even $ q$, and for odd $ q=p^{\alpha}$ with $ \alpha >1$. Later, Smith provided formulas that simplified the computation of $ R(d,q)$ in these cases for $ n>1$. Recently, Cochrane, Liu and Zheng computed upper bounds for $ R(d,q)$ in the general case $ n >0$, stopping short of their explicit evaluation. Here I complete the computation they initiated to obtain explicit values for the Kloosterman sums for $ \alpha >1$, relying on basic properties of some simple specialized exponential sums. The treatment here is more elementary than the author's previous determination of these Kloosterman sums using character theory and $ p$-adic methods. At the least, it provides an alternative, independent evaluation of the Kloosterman sums.

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Additional Information

S. Gurak
Affiliation: Department of Mathematics, University of San Diego, San Diego, California 92110

Received by editor(s): May 10, 2006
Received by editor(s) in revised form: November 8, 2006
Published electronically: May 14, 2007
Dedicated: In memory of Derick H. Lehmer
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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