ON THE GENERAL QUADRATIC GAUSS SUMS WEIGHTED BY CHARACTER SUMS OVER A SHORT INTERVAL
ON THE GENERAL QUADRATIC GAUSS SUMS WEIGHTED BY CHARACTER SUMS OVER A SHORT INTERVAL
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DOI：
10.4134/bkms.2013.50.3.873
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发表时间：
2013-05
影响因子：
0.5
通讯作者：
Tianping Zhang
中科院分区：
文献类型：
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作者：
Tianping Zhang
抽象。利用解析方法研究了短区间上广义二次高斯和按特征和的一次幂均值加权的均值.得到了几个尖锐的渐近公式，表明这些和具有良好的分布性质。此外，它们之间建立了有趣的联系。1.对任意整数n，广义二次高斯和G（n，χ;q）定义为G（n，χ;q）= Xqa =1 χ（a）e纳二q其中e（y）= e2 πiy .这个求和是非常重要的，因为它是经典的二次高斯和的推广。但我们对G（n，χ;q）的性质知之甚少，甚至不知道G（n，χ;q）有多大。因为价值|G（n，χ;q）|当χ变化时是不规则的，人们只能得到一些上界估计。例如，对任意整数n，（n，q）= 1，从科克伦和郑文[1]的一般结果可以推出：|G（n，χ;q）|≤ 2 ω（q）q12，其中ω（q）表示q的不同素因子的个数。q是素数的情况是由于A。Weil [2]，而涉及G（n，χ;q）的加权和[4]具有许多良好的值分布性质，通过这些性质，我们建立了它们之间有趣的联系。现在我们用解析方法研究一般二次高斯和按特征标的一次平均加权的平均值
Abstract. By using the analytic methods, the mean value of the generalquadratic Gauss sums weighted by the first power mean of character sumsover a short interval is investigated. Several sharp asymptotic formulaeare obtained, which show that these sums enjoy good distributive prop-erties. Moreover, interesting connections among them are established. 1. Introduction and main resultsFor any integer n, the general quadratic Gauss sums G(n,χ;q) is defined asG(n,χ;q) =X qa=1 χ(a)ena 2 q,where e(y) = e 2πiy . This summation is very important, since it is the general-ization of the classical quadratic Gauss sums. But we still know little aboutthe properties of G(n,χ;q), we do not even know how large G(n,χ;q) is. Sincethe value of |G(n,χ;q)| is irregular as χ varies, one can only get some upperbound estimates. For example, for any integer n with (n,q) = 1, from thegeneral result of Cochrane and Zheng [1] we can deduce that|G(n,χ;q)| ≤ 2 ω(q) q 12 ,where ω(q) denotes the number of distinct prime divisors of q. The case thatq is prime is due to A. Weil [2].However, weighted sums [4] involving G(n,χ;q) enjoys many good valuedistribution properties, through which interesting connections among them areestablished. Now we shall use analytic methods to study the mean value of thegeneral quadratic Gauss sums weighted by the first power mean of character