Journal of the
Korean Mathematical Society
JKMS

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J. Korean Math. Soc. 2009; 46(4): 733-746

Printed July 1, 2009

https://doi.org/10.4134/JKMS.2009.46.4.733

Copyright © The Korean Mathematical Society.

On the $r$-th hyper-Kloosterman sums and a problem of D. H. Lehmer

Zhang Tianping and Xue Xifeng

Shannxi Normal University and Northwest University

Abstract

For any integer $k\geq 2$, let $P(c,k+1;q)$ be the number of all $k+1$-tuples with positive integer coordinates $(a_1, a_2, \ldots, a_{k+1})$ such that $1\le a_i\le q$, $(a_i, q)=1$, $a_1 a_2\cdots a_{k+1}\equiv c~(\bmod\, q)$ and $2\nmid (a_1+a_2+\cdots+a_{k+1})$, and $E(c,k+1;q)=P(c,k+1;q)-\frac{\phi^{k}(q)}{2}$. The main purpose of this paper is using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet $L$-functions to study the hybrid mean value of the $r$-th hyper-Kloosterman sums $Kl(h,k+1,r;q)$ and $E(c,k+1;q)$, and give an interesting mean value formula.

Keywords: $r$-th hyper-Kloosterman sums, hybrid mean value

MSC numbers: 11L05

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