scholarly journals On the Generalization of Lehmer Problem and High-Dimension Kloosterman Sums

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Guohui Chen ◽  
Han Zhang
Keyword(s):  
High Dimension ◽  
Asymptotic Properties ◽  
Kloosterman Sums ◽  
Analytic Method ◽  
Fixed Integer ◽  
Asymptotic Formulae ◽  
Lehmer Problem

For any fixed integerk≥2and integerrwithr, p=1, it is clear that there existkintegers1≤ai≤p-1 i=1, 2, …, ksuch thata1a2⋯ak≡r mod p. LetN(k,r;p)denote the number of alla1, a2, ⋯aksuch thata1a2⋯ak≡r mod pand 2†a1+a2+⋯ + ak. In this paper, we will use the analytic method and the estimate for high-dimension Kloosterman sums to study the asymptotic properties ofN(k,r;p)and give two interesting asymptotic formulae for it.

scholarly journals On asymptotic properties of the generalized Dirichlet L-functions

2019 ◽  
Vol 15 (06) ◽  
pp. 1305-1321
Author(s):  
Rong Ma ◽  
Yana Niu ◽  
Yulong Zhang
Keyword(s):  
Real Number ◽  
Analytic Continuation ◽  
Asymptotic Properties ◽  
Dirichlet Character ◽  
Mean Value ◽  
Analytic Method ◽  
Asymptotic Formulae ◽  
Number Formula ◽  
The Mean ◽  
Generalized Dirichlet

Let [Formula: see text] be an integer, [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]-functions [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] by analytic continuation. In this paper, we study the mean value properties of the generalized Dirichlet [Formula: see text]-functions, and obtain several sharp asymptotic formulae by using analytic method.

On the Hybrid Mean Value Involving Dedekind Sums and Kloosterman Sums

2015 ◽  
Vol 65 (3) ◽  
Author(s):  
Ma Rong ◽  
Zhang Wenpeng
Keyword(s):  
Asymptotic Properties ◽  
Dirichlet Character ◽  
Mean Value ◽  
Kloosterman Sums ◽  
Dedekind Sums ◽  
Hybrid Mean Value ◽  
Analytic Methods ◽  
Asymptotic Formulae

AbstractThe main purpose of this paper is using the analytic methods and the estimation of Dirichlet character of polynomials to study the asymptotic properties of one kind hybrid mean value involving the Dedekind sums and Kloosterman sums, and give two interesting asymptotic formulae.

scholarly journals GENERALIZED D. H. LEHMER PROBLEM OVER SHORT INTERVALS

2010 ◽  
Vol 53 (2) ◽  
pp. 293-299 ◽  
Author(s):  
PING XI ◽  
YUAN YI
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Asymptotic Properties ◽  
Gauss Sums ◽  
Kloosterman Sums ◽  
Short Intervals ◽  
Lehmer Problem ◽  
Consecutive Integers ◽  
Image Position ◽  
Fixed Positive Integer

AbstractLet n ≥ 2 be a fixed positive integer, q ≥ 3 and c, ℓ be integers with (nc, q)=1 and ℓ|n. Suppose and consist of consecutive integers which are coprime to q. We define the cardinality of a set: The main purpose of this paper is to use the estimates of Gauss sums and Kloosterman sums to study the asymptotic properties of N(, , c, n, ℓ; q), and to give an interesting asymptotic formula for it.

6.—On the Distribution of the Eigenvalues and the Order of the Eigenfunctions of a Fourth-order Singular Boundary Value Problem

1972 ◽  
Vol 71 (1) ◽  
pp. 61-75
Author(s):  
Jyoti Chaudhuri ◽  
W. N. Everitt
Keyword(s):  
Boundary Value Problem ◽  
Real Number ◽  
Boundary Value ◽  
Asymptotic Properties ◽  
Singular Boundary ◽  
Positive Real ◽  
The Real ◽  
Asymptotic Formulae ◽  
Eigenvalue Parameter ◽  
Image Position

SynopsisThis paper is concerned with the asymptotic properties of the eigenvalues and eigenfunctions of the boundary value problemWith suitable restrictions placed on the real-valued coefficient q the spectrum of this problem, with respect to the eigenvalue parameter λ, is discrete; let {λn; n = 1, 2, …} and {ψn; n = 1, 2, …} be the eigenvalues and associated eigenfunctions. Asymptotic formulae are obtained for N(λ), the number of eigenvalues not exceeding the real number λ, and for ψn(x) as n→∞ where x is a fixed, positive real number.

On the Mean Value of a New Arithmetical Function

2012 ◽  
Vol 155-156 ◽  
pp. 396-400
Author(s):  
Ming Jun Wang
Keyword(s):  
Asymptotic Properties ◽  
General Term ◽  
Mean Value ◽  
Arithmetical Function ◽  
Natural Sequence ◽  
Elementary Method ◽  
Hybrid Functions ◽  
Asymptotic Formulae ◽  
The Mean ◽  
Number Theorist

To study one of the problems that Romania number theorist F. Smarandache has proposed and to generalize it. For any positive integer let denotes the natural sequence where each number is repeated times. Based on the general term formula, the asymptotic properties of this sequence and some hybrid functions are studied using the elementary method, the asymptotic formulae are obtained ,thus enriching the study and application of this sequence.

scholarly journals On two sums related to the Lehmer problem over short intervals

2021 ◽  
Vol 6 (11) ◽  
pp. 11723-11732
Author(s):  
Yanbo Song ◽  
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Short Intervals ◽  
Asymptotic Formulae ◽  
Lehmer Problem ◽  
Lehmer Numbers

<abstract><p>In this article, we study sums related to the Lehmer problem over short intervals, and give two asymptotic formulae for them. The original Lehmer problem is to count the numbers coprime to a prime such that the number and the its number theoretical inverse are in different parities in some intervals. The numbers which satisfy these conditions are called Lehmer numbers. It prompts a series of investigations, such as the investigation of the error term in the asymptotic formula. Many scholars investigate the generalized Lehmer problems and get a lot of results. We follow the trend of these investigations and generalize the Lehmer problem.</p></abstract>

scholarly journals The hybrid power mean of quartic Gauss sums and Kloosterman sums

2017 ◽  
Vol 15 (1) ◽  
pp. 151-156 ◽  
Author(s):  
Li Xiaoxue ◽  
Hu Jiayuan
Keyword(s):  
Gauss Sums ◽  
Kloosterman Sums ◽  
Analytic Method ◽  
Computational Formula ◽  
Power Mean ◽  
Computational Problem ◽  
Hybrid Power ◽  
Hybrid Power Mean

Abstract The main purpose of this paper is using the analytic method and the properties of the classical Gauss sums to study the computational problem of one kind fourth hybrid power mean of the quartic Gauss sums and Kloosterman sums, and give an exact computational formula for it.

scholarly journals A mean value related to the D. H. Lehmer problem and Kloosterman sums

2010 ◽  
Vol 143 (3) ◽  
pp. 291-298 ◽  
Author(s):  
Wenpeng Zhang ◽  
Zhaoxia Wu
Keyword(s):  
Mean Value ◽  
Kloosterman Sums ◽  
Lehmer Problem
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