A new hybrid power mean involving the generalized quadratic Gauss sums and sums analogous to Kloosterman sums<Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>

The main purpose of this paper is using the analytic method and properties of the classical quadratic Gauss sums to study the computational problem of a hybrid power mean of generalized quadratic Gauss sums and generalized Kloosterman sums and give an exact computational formula for it.     

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.     

On the fourth power mean of the generalized quadratic Gauss sums

The main purpose of this paper is to use elementary methods and properties of the classical Gauss sums to study the computational problem of one kind of fourth power mean of the generalized quadratic Gauss sums mod q (a positive odd number), and give an exact computational formula for it.     

On character sums over a short interval

The main purpose of this paper is using the analytic methods to study the hybrid mean value involving the character sums, general quadratic Gauss sums and general Kloosterman sums, and give several interesting mean value formulae.     

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.     

On some hybrid power moments of products of generalized quadratic Gauss sums and Kloosterman sums*

In this paper, we investigate hybrid power moments of generalized quadratic Gauss sums weighted with powers of Kloosterman sums and with powers of values of Dirichlet L-functions at 1. We obtain several exact formulas for prime and prime power modulus and some asymptotic formulas.     

两项指数和及两项特征和的混合均值

利用解析方法以及高斯和的性质研究一类二项指数和及二项特征和的混合均值问题,并给出一个精确的表示式.作为应用,给出该和式的一个渐近公式以及该和式与Dirichlet L-函数加权均值的一个较强的渐近公式.     

Explicit formulas for Hecke Gauss sums in quadratic number fields

We derive an explicit formula for Hecke Gauss sums of quadratic number fields. As an immediate consequence we obtain a quadratic reciprocity law in quadratic number fields which generalizes the classical one given by Hecke. The proofs use, apart from the well-known formulas for ordinary Gauss sums, only elementary algebraic manipulations.     

关于Dirichlet L-函数的2k次加权均值

本文的主要目的是利用特征和估计及其三角和方法研究Dirichlet L-函数的2k次加权均值，并得到了一个较为精确的渐近公式。     

广义四次Gauss和的四次均值

本文利用解析方法以及经典Gauss和的性质,研究了模p为奇素数时广义四次Gauss和的四次均值的计算问题,并根据p≡3或1 mod 4,得到了该四次均值的一个精确计算公式和渐近公式.     

On polynomial reciprocity law

The well-known law of quadratic reciprocity has over 150 proofs in print. We establish a relation between polynomial Jacobi symbols and resultants of polynomials over finite fields. Using this relation, we prove the polynomial reciprocity law and obtain a polynomial analogue of classical Burde's quartic reciprocity law. Under the use of our polynomial Poisson summation formula and the evaluation of polynomial exponential map, we get a reciprocity for the generalized polynomial quadratic Gauss sums.     

A Prime Sensitive Hankel Determinant of Jacobi Symbol Enumerators

We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes if and only if n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p − 1 which is the product of a power of p and the generating polynomial of the partial sums of Legendre symbols. The sign of the determinant is determined by the quadratic character of −1 modulo p. The proof of the evaluation makes use of elementary properties of Legendre symbols, quadratic Gauss sums, and orthogonality of trigonometric functions.     

关于Dirichlet L-函数的倒数的偶次幂的加权均值

本文利用特征和与三角和的一些恒等式及其估计式与解析方法讨论了Dirichlet L-函数的倒数的偶次幂的加权均值,得到一个均值分布的渐近公式。     

关于广义$k$次高斯和的均值

The main purpose of this paper is using residue system and character sums methods to investigate the mean value properties of general k-th Gauss sums,and two exact calculating formulas are given.     

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.     

Relative norms of Gauss sums

The explicit determination of the values of Gauss sums is a very classical problem and has some rather deep arithmetic consequences in classfield theory. Here we study the simpler problem of finding their relative norms. We give a complete determination of the relative norms of Gauss sums for norms whose values are known to be in an imaginary quadratic extension of the rational field.     

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     

Mean Value on the Difference Between a Quadratic Residue and Its Inverse Modulo p

The main purpose of this paper is to use the generalized Bernoulli numbers, Gauss sums and the mean value theorems of Dirichlet L-functions between a quadratic residue and its inverse modulo p value formula. to study the asymptotic property of the difference （a prime）, and to give an interesting hybrid mean     

The Fourth Power Mean of the General 2-Dimensional Kloostermann Sums Mod p

The main purpose of this paper is using the analytic methods and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the general 2-dimensional Kloostermann sums mod p, and give an exact computational formula for it.     

The Fourth Power Mean of the General 3-dimensional Kloostermann Sums mod p

The main purpose of this paper is using the analytic methods, the solutions of the congruence equation mod p and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the general 3-dimensional Kloostermann sums mod p, and give a sharp asymptotic formula for it.