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 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 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 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 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 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.

scholarly journals On the Mean Values of Certain Character Sums

2013 ◽  
Vol 2013 ◽  
pp. 1-19
Author(s):  
Zhefeng Xu ◽  
Huaning Liu
Keyword(s):  
Character Sums ◽  
Fourth Power ◽  
Power Mean ◽  
Mean Values ◽  
Asymptotic Formulae ◽  
Fourth Power Mean ◽  
The Mean

Letq≥5be an odd number. In this paper, we study the fourth power mean of certain character sums∑χmodq,χ-1=-1*∑1≤a≤q/4aχa4and∑χmodq,χ-1=1*∑1≤a≤q/4aχa4, where∑‍*denotes the summation over primitive characters moduloq, and give some asymptotic formulae.

On the Fourth Power Mean of the Character Sums Over Short Intervals

2006 ◽  
Vol 23 (1) ◽  
pp. 153-164 ◽  
Author(s):  
Wen Peng Zhang ◽  
Xiao Ying Wang
Keyword(s):  
Character Sums ◽  
Fourth Power ◽  
Power Mean ◽  
Short Intervals ◽  
Fourth Power Mean

scholarly journals A certain new Gauss sum and its fourth power mean

2020 ◽  
Vol 5 (5) ◽  
pp. 5004-5011
Author(s):  
Yan Zhao ◽  
◽  
Wenpeng Zhang ◽  
Xingxing Lv ◽  
Keyword(s):  
Gauss Sum ◽  
Fourth Power ◽  
Power Mean ◽  
Fourth Power Mean

On the fourth power mean of the generalized quadratic Gauss sums

2017 ◽  
Vol 34 (6) ◽  
pp. 1037-1049 ◽  
Author(s):  
Wen Peng Zhang ◽  
Xin Lin
Keyword(s):  
Gauss Sums ◽  
Fourth Power ◽  
Power Mean ◽  
Fourth Power Mean

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].

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