scholarly journals One Kind Special Gauss Sums and their Mean Square Values

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jianhong Zhao ◽  
Jiejie Gao
Keyword(s):  
Mean Value ◽  
Gauss Sums ◽  
Mean Square ◽  
Analytic Methods ◽  
The Mean

In this paper, we introduce one kind special Gauss sums; then, using the elementary and analytic methods to study the mean value properties of these kind sums, we obtain several exact calculating formulae for them.

scholarly journals On a sum analogous to Dedekind sum and its mean square value formula

2002 ◽  
Vol 32 (1) ◽  
pp. 47-55 ◽  
Author(s):  
Zhang Wenpeng
Keyword(s):  
Asymptotic Property ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Mean Square ◽  
Dedekind Sum ◽  
The Mean

The main purpose of this paper is using the mean value theorem of DirichletL-functions to study the asymptotic property of a sum analogous to Dedekind sum, and give an interesting mean square value formula.

scholarly journals The hybrid mean value of Dedekind sums and two-term exponential sums

2016 ◽  
Vol 14 (1) ◽  
pp. 436-442
Author(s):  
Chang Leran ◽  
Li Xiaoxue
Keyword(s):  
Asymptotic Formula ◽  
Exponential Sums ◽  
Mean Value ◽  
Gauss Sums ◽  
Mean Value Theorem ◽  
Dedekind Sums ◽  
Hybrid Mean Value ◽  
The Mean ◽  
Interesting Identity

AbstractIn this paper, we use the mean value theorem of Dirichlet L-functions, the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the two-term exponential sums, and give an interesting identity and asymptotic formula for it.

scholarly journals On some mean value results involving djζ(1/2 + it)dj

2003 ◽  
Vol 127 (28) ◽  
pp. 17-29
Author(s):  
Aleksandar Ivic
Keyword(s):  
Mean Value ◽  
Fourth Moment ◽  
Mean Square ◽  
Error Terms ◽  
The Mean

Several problems involving E(T) and E2(T), the error terms in the mean square and mean fourth moment formula for |?(1/2 + it)|, are discussed. In particular it is proved that ?0T? E(t)E2(T)dt?T7/4(logT)7/2loglogT. .

scholarly journals Failure Probability Analysis of Interlayer Sliding Belts of Gravity Dams Under Seismic Load

2014 ◽  
Vol 8 (1) ◽  
pp. 85-91 ◽  
Author(s):  
Qiang Xu ◽  
Jian-Yun Chen ◽  
Chunfeng Zhao ◽  
Jing Li ◽  
Hongyuan Yue
Keyword(s):  
Failure Probability ◽  
Mean Value ◽  
Seismic Load ◽  
Mean Square ◽  
Mean Square Deviation ◽  
Concrete Gravity Dam ◽  
Gravity Dams ◽  
The Mean ◽  
Improved Model ◽  
Pseudo Excitation

In this paper, an improved model is presented for analysis of failure probability of the interlayer sliding belts under seismic loads. Firstly, using the theory of the Markov chain, the relation between failure probabilities of specified interlayer sliding belts and elements in this interlayer sliding belt is deduced. Then, the failure function is proposed according to the destructive characteristic of concrete, the pseudo excitation method is utilized in order to obtain the probability distribution of element stresses in specified interlayer sliding belts, and the improved response surface method based on weighted regression is used to calculate the failure probability of elements in specified interlayer sliding belts. Finally, an algorithm is established to calculate the failure probability of the specified interlayer sliding belts. In this paper, the mean value and variance of the tensile strength of elements are changed when interlayer sliding belt is developed. The numerical results show that the conditional failure probability in specified interlayer sliding belts at the head of the dam tends to decrease. However, the tendency of conditional failure probability in the other specified interlayer sliding belts is complicated. And the interlayer sliding belt at head of the concrete gravity dam is the most dangerous. In addition, the tendencies of the mean value and mean square deviation of stresses in different specified interlayer sliding belts are similar. The mean value of stress in different specified interlayer sliding belts tends to decrease but the tendency of mean square deviation changes from decrease to increase. The range of the mean value and the mean square deviation of stress in specified interlayer sliding belts at the heel of the dam is the greatest of all.

scholarly journals The mean value of the generalized Dirichlet L-functions with the weight of the Gauss sums

2021 ◽  
pp. 1-18
Author(s):  
Yana Niu ◽  
Rong Ma ◽  
Yulong Zhang ◽  
Peilin Jiang
Keyword(s):  
Asymptotic Formula ◽  
Real Number ◽  
Analytic Continuation ◽  
Dirichlet Character ◽  
Mean Value ◽  
Gauss Sums ◽  
Number Formula ◽  
Sum Formula ◽  
The Mean ◽  
Generalized Dirichlet

Let [Formula: see text] be an integer, and let [Formula: see text] denote a Dirichlet character modulo [Formula: see text]. For any real number [Formula: see text], we define the generalized Dirichlet [Formula: see text]-function as [Formula: see text] where [Formula: see text] with [Formula: see text] and [Formula: see text] both real. It can be extended to all [Formula: see text] using analytic continuation. For any integer [Formula: see text], the famous Gauss sum [Formula: see text] is defined as [Formula: see text] where [Formula: see text]. This paper uses analytic methods to study the mean value properties of the generalized Dirichlet [Formula: see text]-functions with the weight of the Gauss sums, and a sharp asymptotic formula is obtained.

scholarly journals Meteorological Temperature and Humidity Prediction from Fourier-Statistical Analysis of Hourly Data

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Alejandro Castañeda-Miranda ◽  
M. de Icaza-Herrera ◽  
Víctor M. Castaño
Keyword(s):  
Angular Position ◽  
Geographical Location ◽  
Mean Value ◽  
Mean Square ◽  
Mean Square Deviation ◽  
Hourly Data ◽  
Humidity Measurements ◽  
The Mean ◽  
Length Of The Day ◽  
New Variables

The temperature readings for all the 365 days and the 24 hours may be fitted through a 3 × 3 matrix (the so-called T-matrix). The mean square deviation between this fit and the actual meteorological measurements is smaller than three degrees Celsius. Four entries of this (nonsymmetric) matrix may be fixed by other means, leaving only five independent components. However, the same method applied to the humidity measurements produces a larger mean square deviation. A strong stochastical connection is found between the T-temperature matrix and the U-humidity matrix. The computer program, in C, may be used to adjust a (2M + 1) × (2m + 1) matrix simply by changing the arguments at the command line and has been tested with m and M ranging from zero to 11 (eleven) (more than 24 readings per day are necessary for larger values of m). The physical meaning of these constants is given only in the case m = M = 1. Our results have also been connected to fundamental cosmological properties: Earth’s orbit, the ecliptic angle, and the latitude of Querétaro (or whatever geographical location is chosen). A separate program calculates the angular position of the Sun as measured in the sky of Querétaro, to determine the length of the day or the mean value of the solar cosine. This work introduces several new variables which happen to be stochastically connected.

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 On the General Dedekind Sums and Two-Term Exponential Sums

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Junli Zhang ◽  
Wenpeng Zhang
Keyword(s):  
Exponential Sums ◽  
Mean Value ◽  
Gauss Sums ◽  
Dedekind Sums ◽  
Computational Formula ◽  
Computational Problem ◽  
Hybrid Mean Value ◽  
Analytic Methods

We use the analytic methods and the properties of Gauss sums to study the computational problem of one kind hybrid mean value involving the general Dedekind sums and the two-term exponential sums, and give an interesting computational formula for it.

The mean-square value of the divisor function

2016 ◽  
Vol 100 (548) ◽  
pp. 203-212
Author(s):  
Peter Shiu
Keyword(s):  
Counting Function ◽  
Mean Value ◽  
Mean Square ◽  
Divisor Function ◽  
Dirichlet’S Theorem ◽  
Fixed Power ◽  
The Mean ◽  
Dirichlet's Theorem ◽  
Image Position ◽  
Prime Counting Function

The behaviour of the divisor function d (n) is rather tricky. For a prime p, we have d(p) = 2, but if n is the product of the first k primes then, by Chebyshev's estimate for the prime counting function [1, Theorem 414], we have so thatfor such n then, d (n) is ‘unusually large’ — it can exceed any fixed power of log n, for example.In [2] Jameson gives, amongst other things, a derivation of Dirichlet's theorem, which shows that the mean-value of the divisor function in an interval containing n is log n. However, the result is somewhat deceptive because, for most n, the value of d (n) is substantially smaller than log n.

Sign in / Sign up

Export Citation Format

Share Document