COMPORTEMENT AU VOISINAGE DE 1 DE LA FONCTION DE RÉPARTITION DE φ(n)/n

2009 ◽  
Vol 05 (08) ◽  
pp. 1347-1384 ◽  
Author(s):  
VINCENT TOULMONDE
Keyword(s):  
Differential Equation ◽  
Distribution Function ◽  
Asymptotic Expansion ◽  
Functional Equations ◽  
Error Term ◽  
Quantitative Measure ◽  
Prime Number Theorem ◽  
Nous Obtenons ◽  
Prime Factors ◽  
Equation Différentielle

Let φ denote Euler's totient function, and G be the distribution function of φ(n)/n. Using functional equations, it is shown that φ(n)/n is statistically close to 1 essentially when prime factors of n are large. A function defined by a difference-differential equation gives a quantitative measure of the statistical influence of the size of prime factors of n on the closeness of φ(n)/n to 1. As a corollary, an asymptotic expansion at any order of G(1)-G(1-ε) is obtained according to negative powers of log (1/ε), when ε tends to 0+. This improves a result of Erdős (1946) in which he gives the first term of it. By optimally choosing the order of this expansion, an estimation of G(1)-G(1-ε) is deduced, involving an error term of the same size as the best known error term involved in prime number theorem. Soit φ l'indicatrice d'Euler. Nous étudions le comportement au voisinage de 1 de la fonction G de répartition de φ(n)/n, via la mise en évidence d'équations fonctionnelles. Nous obtenons un résultat mesurant l'influence statistique de la taille du plus petit facteur premier d'un entier générique n quant à la proximité de φ(n)/n par rapport à 1. Ce résultat met en jeu une fonction définie par une équation différentielle aux différences. Nous en déduisons un développement limité à tout ordre de G(1)-G(1-ε) selon les puissances de 1/(log 1/ε), améliorant ainsi un résultat d'Erdős (1946) dans lequel il obtient le premier terme de ce développement. Une troncature convenable de ce développement fournit un terme d'erreur comparable à celui actuellement connu pour le théorème des nombres premiers.

scholarly journals Integers free of small prime factors in arithmetic progressions

2000 ◽  
Vol 157 ◽  
pp. 103-127 ◽  
Author(s):  
Ti Zuo Xuan
Keyword(s):  
Prime Number ◽  
Short Range ◽  
Error Term ◽  
Real Zero ◽  
Prime Number Theorem ◽  
Arithmetic Progressions ◽  
Real Character ◽  
Dickman Function ◽  
Prime Factors ◽  
Positive Integers

For real x ≥ y ≥ 2 and positive integers a, q, let Φ(x, y; a, q) denote the number of positive integers ≤ x, free of prime factors ≤ y and satisfying n ≡ a (mod q). By the fundamental lemma of sieve, it follows that for (a,q) = 1, Φ(x,y;a,q) = φ(q)-1, Φ(x, y){1 + O(exp(-u(log u- log2 3u- 2))) + (u = log x log y) holds uniformly in a wider ranges of x, y and q.Let χ be any character to the modulus q, and L(s, χ) be the corresponding L-function. Let be a (‘exceptional’) real character to the modulus q for which L(s, ) have a (‘exceptional’) real zero satisfying > 1 - c0/log q. In the paper, we prove that in a slightly short range of q the above first error term can be replaced by where ρ(u) is Dickman function, and ρ′(u) = dρ(u)/du.The result is an analogue of the prime number theorem for arithmetic progressions. From the result can deduce that the above first error term can be omitted, if suppose that 1 < q < (log q)A.

scholarly journals Prime geodesics and averages of the Zagier L-series

2021 ◽  
pp. 1-24
Author(s):  
OLGA BALKANOVA ◽  
DMITRY FROLENKOV ◽  
MORTEN S. RISAGER
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Error Term ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Prime Geodesic Theorem ◽  
Error Terms ◽  
Average Size ◽  
Real Quadratic

Abstract The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

An Inventory With Constant Demand and Poisson Restocking

1987 ◽  
Vol 1 (2) ◽  
pp. 203-210 ◽  
Author(s):  
Laurence A. Baxter ◽  
Eui Yong Lee
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Distribution Function ◽  
Poisson Process ◽  
Stationary Case ◽  
Stieltjes Transform ◽  
Partial Differential

An inventory whose stock decreases linearly with time is considered. The inventory may be replenished at the instants at which a deliveryman arrives provided that the level of the inventory does not exceed a certain threshold; deliveries are made according to a Poisson process. A partial differential equation for the distribution function of the level of the inventory is solved to yield a formula for the corresponding Laplace–Stieltjes transform. The evaluation of the transform is discussed and explicit results are obtained for the stationary case.

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

ESTIMATES FOR THE ERROR TERM IN A UNIFORM ASYMPTOTIC EXPANSION OF THE JACOBI POLYNOMIALS

2003 ◽  
Vol 01 (02) ◽  
pp. 213-241 ◽  
Author(s):  
R. WONG ◽  
Y.-Q. ZHAO
Keyword(s):  
Asymptotic Expansion ◽  
Explicit Expression ◽  
Canonical Form ◽  
Error Term ◽  
Jacobi Polynomials ◽  
Recursive Formula ◽  
Contour Integral ◽  
Integration By Parts ◽  
Uniform Asymptotic Expansion ◽  
Repeated Use

There are now several ways to derive an asymptotic expansion for [Formula: see text], as n → ∞, which holds uniformly for [Formula: see text]. One of these starts with a contour integral, involves a transformation which takes this integral into a canonical form, and makes repeated use of an integration-by-parts technique. There are two advantages to this approach: (i) it provides a recursive formula for calculating the coefficients in the expansion, and (ii) it leads to an explicit expression for the error term. In this paper, we point out that the estimate for the error term given previously is not sufficient for the expansion to be regarded as genuinely uniform for θ near the origin, when one takes into account the behavior of the coefficients near θ = 0. Our purpose here is to use an alternative method to estimate the remainder. First, we show that the coefficients in the expansion are bounded for [Formula: see text]. Next, we give an estimate for the error term which is of the same order as the first neglected term.

scholarly journals INFLUENCE OF INCONSTANT DISTRIBUTION OF DEADWOOD BEARING STIFFNESS COEFFICIENTON NATURAL FREQUENCY OF THE SHAFT TRANSVERSE VIBRATIONS

2019 ◽  
pp. 89-96
Author(s):  
Guriy Kushner ◽  
Victor Andreevich Mamontov
Keyword(s):  
Differential Equation ◽  
Distribution Function ◽  
Uniform Distribution ◽  
Natural Frequency ◽  
Stiffness Coefficient ◽  
Propeller Shaft ◽  
Shaft Length ◽  
Slide Bearing ◽  
Transverse Vibrations ◽  
Stiffness Distribution

One of the most significant factors affecting the natural frequency of transverse vibrations of shaft-slide bearing systems is the stiffness coefficient of the slide bearing. The need to consider the influence of heterogeneity of stiffness coefficient of the bearing on its natural frequency is caused by the fact that when the bearing is worn, the modulus of longitudinal elasticity of the material changes, and since the bearing wears unevenly, the non-uniform distribution of the stiffness coefficient occurs. The problem of determining the natural frequency of transverse vibrations of a ship propeller shaft based on the foundation with a variable stiffness coefficient along the length has been studied. The differential equation of the propeller shaft vibrations written taking into account the stiffness coefficient variable along the shaft length is considered. It has been noted that, in the general case, this equation is a fourth-order partial differential equation and cannot be integrated in quadratures for an arbitrary stiffness distribution function along the shaft length. A numerical-analytical method for determining the natural frequency of a system based on approximation of the stiffness distribution function by a piecewise-linear function is proposed. The method is applied to calculate the natural frequencies of the pipeline section taking into account the functions describing the change in the stiffness coefficient. The proposed method allows to consider the section of the shafting enclosed in the stern bearing, subject to the non-uniform distribution of the stiffness coefficient of the bearing, and is the basis for improving the accuracy of finding the true natural frequency of transverse vibrations of the shafting.

scholarly journals Quantitative measure of hysteresis for memristors through explicit dynamics

2012 ◽  
Vol 468 (2144) ◽  
pp. 2210-2229 ◽  
Author(s):  
P. S. Georgiou ◽  
S. N. Yaliraki ◽  
E. M. Drakakis ◽  
M. Barahona
Keyword(s):  
Differential Equation ◽  
Analytical Solutions ◽  
Quantitative Measure ◽  
Explicit Solutions ◽  
Mathematical Framework ◽  
Input Output ◽  
Explicit Formulas ◽  
Hewlett Packard ◽  
Lumped Parameter ◽  
Proposed Model

We introduce a mathematical framework for the analysis of the input–output dynamics of externally driven memristors. We show that, under general assumptions, their dynamics comply with a Bernoulli differential equation and hence can be nonlinearly transformed into a formally solvable linear equation. The Bernoulli formalism, which applies to both charge- and flux-controlled memristors when either current or voltage driven, can, in some cases, lead to expressions of the output of the device as an explicit function of the input. We apply our framework to obtain analytical solutions of the i – v characteristics of the recently proposed model of the Hewlett–Packard memristor under three different drives without the need for numerical simulations. Our explicit solutions allow us to identify a dimensionless lumped parameter that combines device-specific parameters with properties of the input drive. This parameter governs the memristive behaviour of the device and, consequently, the amount of hysteresis in the i – v . We proceed further by defining formally a quantitative measure for the hysteresis of the device, for which we obtain explicit formulas in terms of the aforementioned parameter, and we discuss the applicability of the analysis for the design and analysis of memristor devices.

Asymptotic Expansion for a Class of Sample Quantiles

1982 ◽  
Vol 31 (1-2) ◽  
pp. 1-11
Author(s):  
Kamal C. Chanda
Keyword(s):  
Distribution Function ◽  
Asymptotic Expansion ◽  
Present Article ◽  
Random Sample ◽  
Order Statistic ◽  
Regularity Conditions ◽  
Real Numbers ◽  
Positive Integers ◽  
Sample Quantiles

Let X k: n be the kth order statistic (1 ⩽ k ⩽ n) for a random sample of size n from a population with the distribution function F. Let {α n}, { βn} ( βn > 0) be sequences of real numbers and let { kn} ( kn ⩽ n) be a sequence of positive integers. The present article explores the various choices of α n, βn and kn such that under some mild regularity conditions on F, L( Yn) → n (0,1) as n→∞, where Yn = ( Xkn:n + α n)⁄ βn. It is further shown that under some additional conditions on F, standard asymptotic expansion (in Edgeworth form) for the distribution of Yn can be derived.

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