On an identity associated with Weil's estimate and its applications

In this paper we study the fourth mean value of generalized quadratic Gauss sums, and get an interesting identity associated with Weil's estimate,

     

The First Power Mean of the Inversion of L-Functions Weighted by Quadratic Gauss Sums

The main purpose of this paper is to use estimates for character sums and analytic methods to study the first power mean of the inversion of Dirichlet L-functions with the weight of general quadratic Gauss sums, and three asymptotic formulae are obtained.     

二次剩余问题的一个新定理

研究了二次剩余问题,利用整数分类的办法,给出了|Jn|和|Qn|的公式(这里n是奇合数,Jn是Zn*中有Jacobi符号为1的所有元素的集合,Qn是模n的所有二次剩余的集合) .基于这些结果,可以得出当a对模n的Jacobi符号等于1时,正确猜测a为模n的二次剩余的可能性,从而推广了[1]p .74中的结果.     

A note on the Cochrane sum and its hybrid mean value formula

In this paper, we use the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of Cochrane sums and general Kloosterman sums, and give two sharp asymptotic formulae.     

关于特征和的混合均值

研究了特征和及其算术性质,并利用解析方法得到了两个关于特征和的混合均值的渐近公式,从而推广了特征和的算术性质。     

Gauss sums and multinomial coefficients

Consider a Gauss sum for a finite field of characteristic p, where p is an odd prime. When such a sum (or a product of such sums) is a p-adic integer we show how it can be realized as a p-adic limit of a sequence of multinomial coefficients. As an application we generalize some congruences of Hahn and Lee to exhibit p-adic limit formulae, in terms of multinomial coefficients, for certain algebraic integers in imaginary quadratic fields related to the splitting of rational primes. We also give an example illustrating how such congruences arise from a p-integral formal group law attached to the p-adic unit part of a product of Gauss sums.     

关于Dirichlet L-函数的一次加权均值

利用三角和估计及其解析方法研究DirichletL函数的一次加权均值,得到了一个较为精确的渐近公式.     

An Application of Exponential Sum Estimates

Let p be an odd prime and let δ be a fixed real number with 0<δ<2. For an integer a with 0<a<p, denote by ā the unique integer between 0 and p satisfying a ā≡1 (mod p). Further, let {x} denote the fractional part of x. We derive an asymptotic formula for the number of pairs of integers (a, b with . This work is supported by N. S. F. of P. R. China (10271093)     

关于广义Dedekind和的2k次均值

本文利用 Ｄｉｒｉｃｈｌｅｔ Ｌ－函数的均值定理,研究广义 Ｄｅｄｅｋｉｎｄ和的渐近性质,并给出一个有趣的渐近公式．     

Moments of Generalized Quadratic Gauss Sums Weighted by L-Functions

The main purpose of this paper is using estimates for character sums and analytic methods to study the second, fourth, and sixth order moments of generalized quadratic Gauss sums weighted by L-functions. Three asymptotic formulae are obtained.     

关于三次高斯和的均值

主要研究三次高斯和的均值性质,并给出一个较强的渐近公式。     

On the 2<Emphasis Type="Italic">k</Emphasis>-th Power Mean of Inversion of <Emphasis Type="Italic">L</Emphasis>-functions with the Weight of the Gauss Sum

Abstract The main purpose of this paper is to use the estimate for character sums and the method of trigonometric sums to study the 2k-th power mean of the inversion of Dirichlet L-functions with the weight of the Gauss sums, and give a sharper asymptotic formula. This work is supported by the Doctorate Foundation of Xi’an Jiaotong University     

Incomplete higher-order Gauss sums

We consider the classical incomplete higher-order Gauss sums

     

A hybrid mean value formula involving Kloosterman sums and Hurwitz zeta-function

The main purpose of this paper is using the mean value theorem of Dirichlet L-function and the estimates for character sums to study the asymptotic properties of a hybrid mean value of Kloosterman sums with the weight of Hurwitz zeta-function and the Cochrane sums, and give an interesting mean value formula for it.     

On the general Kloosterman sum and its fourth power mean

The main purpose of this paper is to study the asymptotic property of the fourth power mean of the general Kloosterman sums, and give an interesting calculating formula.     

三次高斯和与Kloosterman和的线性递推公式

应用三角和方法以及高斯和的若干性质,研究三次高斯和与Kloosterman和的一类高次混合均值的计算问题,本文给出该混合均值的一个有趣的线性递推公式.同时,还应用该递推公式,得到三次高斯和与Kloosterman和的高次混合均值的一系列较强的渐近公式.     

关于Dedekind和的均值公式

本文的主要目的是利用ＤｉｒｉｃｈｌｅｔＬ－函数的均值定理研究Ｄｅｄｅｋｉｎｄ和的值分布性质,并给出一个较强的均值公式．     

On the mean value formula of dedekind sums

The main purpose of this paper is to use the mean value theorem of the Dirichlet L-function to study the distribution property of Dedekind sums, and to give a sharper mean value formula. This work is supported by the National Natural Science Foundation of P. R. China