一个特殊的Gauss和以及它的上界估计

A Special Gauss Sums and Its Upper Bound Estimate

设p是一个奇素数.对任意满足1≤a≤p-1的整数a,存在唯一的整数1≤a≤p-1,使得a·a≡1 mod p成立.设N(p)表示区间1≤a≤p-1中所有满足条件a与a具有相反奇偶性的a的集合.本文利用解析方法以及广义Kloosterman和的性质研究一类特殊的Gauss和∑。∈N(p)x(a)e(ma/p)的估计问题,给出一个较强的上界估计,其中e(x)=e^(2πix),(m,p)=1,且x是模p的任意特征.

Let p be an odd prime.For each integer a with 1≤ a ≤ p- 1,we know that there exists one and only one a with 0≤ a ≤ p- 1 such that a· a ≡1 mod p.Let N(p) denote the set of all integers 1≤a≤p-1 in which a and a are of opposite parity.The main purpose of this paper is using the analytic method and the properties of general Kloosterman sums to study the estimate problem of the Gauss sums ∑a∈N(p) x(a)e(ma/p),and give a sharp upper bound estimate for it,where e(x)=e2πix,(m,p) = 1,and x denotes any character mod p.