scholarly journals A New Sum Analogous to Gauss Sums and Its Fourth Power Mean

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Shaofeng Ru ◽  
Wenpeng Zhang
Keyword(s):  
Upper Bound ◽  
Gauss Sums ◽  
Fourth Power ◽  
Power Mean ◽  
Computational Problem ◽  
Analytic Methods ◽  
Fourth Power Mean ◽  
Upper Bound Estimate

The main purpose of this paper is to use the analytic methods and the properties of Gauss sums to study the computational problem of one kind of new sum analogous to Gauss sums and give an interesting fourth power mean and a sharp upper bound estimate for it.

scholarly journals On the Fourth Power Mean of the Two-Term Exponential Sums

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Han Zhang ◽  
Wenpeng Zhang
Keyword(s):  
Asymptotic Formula ◽  
Exponential Sums ◽  
Gauss Sums ◽  
Fourth Power ◽  
Power Mean ◽  
Computational Problem ◽  
Analytic Methods ◽  
Fourth Power Mean ◽  
Interesting Identity

The main purpose of this paper is to use the analytic methods and the properties of Gauss sums to study the computational problem of one kind fourth power mean of two-term exponential sums and give an interesting identity and asymptotic formula for it.

scholarly journals The Generalized Quadratic Gauss Sum and Its Fourth Power Mean

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 258
Author(s):  
Shimeng Shen ◽  
Wenpeng Zhang
Keyword(s):  
Character Sums ◽  
Gauss Sums ◽  
Gauss Sum ◽  
Fourth Power ◽  
Power Mean ◽  
Analytic Methods ◽  
Fourth Power Mean

In this article, our main purpose is to introduce a new and generalized quadratic Gauss sum. By using analytic methods, the properties of classical Gauss sums, and character sums, we consider the calculating problem of its fourth power mean and give two interesting computational formulae for it.

scholarly journals A new fourth power mean of two-term exponential sums

2019 ◽  
Vol 17 (1) ◽  
pp. 407-414
Author(s):  
Chen Li ◽  
Wang Xiao
Keyword(s):  
Asymptotic Formula ◽  
Exponential Sums ◽  
Gauss Sums ◽  
Fourth Power ◽  
Power Mean ◽  
Analytic Methods ◽  
Fourth Power Mean

Abstract The main purpose of this paper is to use analytic methods and properties of quartic Gauss sums to study a special fourth power mean of a two-term exponential sums modp, with p an odd prime, and prove interesting new identities. As an application of our results, we also obtain a sharp asymptotic formula for the fourth power mean.

scholarly journals New Mixed Exponential Sums and Their Application

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Yu Zhan ◽  
Xiaoxue Li
Keyword(s):  
Asymptotic Formula ◽  
Upper Bound ◽  
Exponential Sums ◽  
Mean Value ◽  
Fourth Power ◽  
Computational Formula ◽  
Power Mean ◽  
Fourth Power Mean ◽  
Upper Bound Estimate ◽  
The Mean

The main purpose of this paper is to introduce a new mixed exponential sums and then use the analytic methods and the properties of Gauss sums to study the computational problems of the mean value involving these sums and give an interesting computational formula and a sharp upper bound estimate for these mixed exponential sums. As an application, we give a new asymptotic formula for the fourth power mean of DirichletL-functions with the weight of these mixed exponential sums.

scholarly journals The Recursive Properties of the Error Term of the Fourth Power Mean of the Generalized Cubic Gauss Sums

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Zhang Jin ◽  
Zhang Jiafan
Keyword(s):  
Error Term ◽  
Gauss Sums ◽  
Fourth Power ◽  
Power Mean ◽  
Analytic Methods ◽  
Fourth Power Mean ◽  
Recurrence Formulae

In this paper, we use the analytic methods and the properties of the classical Gauss sums to study the properties of the error term of the fourth power mean of the generalized cubic Gauss sums and give two recurrence formulae for it.

On the mean value of mixed exponential sums

2016 ◽  
Vol 13 (06) ◽  
pp. 1515-1530
Author(s):  
Ming-Liang Gong ◽  
Ya-Li Li
Keyword(s):  
Explicit Formula ◽  
Exponential Sums ◽  
Dirichlet Character ◽  
Mean Value ◽  
Fourth Power ◽  
Power Mean ◽  
Analytic Methods ◽  
Fourth Power Mean ◽  
The Mean

We use analytic methods to obtain an explicit formula for the fourth power mean [Formula: see text] where [Formula: see text], [Formula: see text] is a Dirichlet character modulo [Formula: see text] and [Formula: see text] denotes the summation over all [Formula: see text] such that [Formula: see text]. This extends the result of Chen, Ai and Cai by overcoming the limitation [Formula: see text].

scholarly journals Dirichlet characters of the rational polynomials

2021 ◽  
Vol 7 (3) ◽  
pp. 3494-3508
Author(s):  
Wenjia Guo ◽  
◽  
Xiaoge Liu ◽  
Tianping Zhang
Keyword(s):  
Dirichlet Character ◽  
Character Sums ◽  
Gauss Sums ◽  
Fourth Power ◽  
Power Mean ◽  
Dirichlet Characters ◽  
Fourth Power Mean ◽  
Rational Polynomials

<abstract><p>Denote by $ \chi $ a Dirichlet character modulo $ q\geq 3 $, and $ \overline{a} $ means $ a\cdot\overline{a} \equiv 1 \bmod q $. In this paper, we study Dirichlet characters of the rational polynomials in the form</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \sum\limits^{q}_{a = 1}'\chi(ma+\overline{a}), $\end{document} </tex-math></disp-formula></p> <p>where $ \sum\limits_{a = 1}^{q}' $ denotes the summation over all $ 1\le a\le q $ with $ (a, q) = 1 $. Relying on the properties of character sums and Gauss sums, we obtain W. P. Zhang and T. T. Wang's identity <sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup> under a more relaxed situation. We also derive some new identities for the fourth power mean of it by adding some new ingredients.</p></abstract>

scholarly journals On the two-term exponential sums and character sums of polynomials

2019 ◽  
Vol 17 (1) ◽  
pp. 1239-1248
Author(s):  
Yuankui Ma ◽  
Wenpeng Zhang
Keyword(s):  
Exponential Sums ◽  
Character Sums ◽  
Gauss Sums ◽  
Asymptotic Formulas ◽  
Power Mean ◽  
Computational Problem ◽  
Hybrid Power ◽  
Analytic Methods ◽  
Hybrid Power Mean

Abstract The main aim of this paper is to use the analytic methods and the properties of the classical Gauss sums to research the computational problem of one kind hybrid power mean containing the character sums of polynomials and two-term exponential sums modulo p, an odd prime, and acquire several accurate asymptotic formulas for them.

scholarly journals On the Hybrid Power Mean Involving the Character Sums and Dedekind Sums

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Xiaoling Xu
Keyword(s):  
Character Sums ◽  
Gauss Sums ◽  
Dedekind Sums ◽  
Character Sum ◽  
Kloosterman Sum ◽  
Power Mean ◽  
Computational Problem ◽  
Hybrid Power ◽  
Analytic Methods ◽  
Hybrid Power Mean

The main purpose of this paper is to use the elementary and analytic methods, the properties of Gauss sums, and character sums to study the computational problem of a certain hybrid power mean involving the Dedekind sums and a character sum analogous to Kloosterman sum and give two interesting identities for them.

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