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.
Zhang Tianping and Xue Xifeng
Shannxi Normal University and Northwest University
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
2006; 43(6): 1199-1217
2010; 47(6): 1107-1122
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd