Error term estimate of order n?2 for the distribution function of Smirnov's two-sample statistic

1988 ◽  
Vol 40 (2) ◽  
pp. 214-220
Author(s):  
A. I. Orlov ◽  
I. V. Orlovskii
Keyword(s):  
Distribution Function ◽  
Error Term ◽  
Sample Statistic

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 Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers

2020 ◽  
Author(s):  
Giordano Cotti
Keyword(s):  
Distribution Function ◽  
Analytic Continuation ◽  
Dirichlet Series ◽  
Riemann Hypothesis ◽  
Error Term ◽  
Asymptotic Estimate ◽  
Quantum Cohomology ◽  
Prime Numbers ◽  
Canonical Coordinates ◽  
Small Quantum

Abstract The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of small quantum cohomology of complex Grassmannians are studied. It is shown that surprisingly this frequency is strictly subordinate and highly influenced by the distribution of prime numbers. Two equivalent formulations of the Riemann hypothesis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associated to the sequence counting non-coalescing Grassmannians, and the latter as asymptotic estimate (whose error term cannot be improved) for their distribution function.

scholarly journals The Existence of a Distribution Function for an Error Term Related to the Euler Function

1955 ◽  
Vol 7 ◽  
pp. 63-75 ◽  
Author(s):  
Paul Erdös ◽  
H. N. Shapiro
Keyword(s):  
Distribution Function ◽  
Error Term ◽  
Average Order ◽  
Euler Function ◽  
Image Position

The average order of the Euler function ϕ(n), the number of integers less than n which are relatively prime to n, raises many difficult and still unanswered questions. Thus, for1.1,and1.2,it is known that R(x) = O(x log x) and H(x) = O(log x). However, though these results are quite old, they were not improved until recently.

scholarly journals Issues and Strategies for Aggregate Supply Response Estimation for Policy Analyses

2004 ◽  
Vol 36 (2) ◽  
pp. 351-367
Author(s):  
Octavio A. Ramirez ◽  
Samarendu Mohanty ◽  
Carlos E. Carpio ◽  
Megan Denning
Keyword(s):  
Distribution Function ◽  
Confidence Intervals ◽  
Probability Distribution Function ◽  
Error Term ◽  
Southeastern United States ◽  
Small Sample ◽  
Supply Response ◽  
Random Components ◽  
Standard Models ◽  
To Come

We demonstrate the use of the small-sample econometrics principles and strategies to come up with reliable yield and acreage models for policy analyses. We focus on demonstrating the importance of proper representation of systematic and random components of the model for improving forecasting precision along with more reliable confidence intervals for the forecasts. A probability distribution function modeling approach, which has been shown to provide more reliable confidence intervals for the dependent variable forecasts than the standard models that assume error term normality, is used to estimate cotton supply response in the Southeastern United States.

scholarly journals Effective limit distribution of the Frobenius numbers

2014 ◽  
Vol 151 (5) ◽  
pp. 898-916 ◽  
Author(s):  
Han Li
Keyword(s):  
Distribution Function ◽  
Linear Combination ◽  
Asymptotic Distribution ◽  
Limit Distribution ◽  
Lattice Point ◽  
Error Term ◽  
Smooth Boundary ◽  
Piecewise Smooth Boundary ◽  
Frobenius Numbers ◽  
Invent Math

The Frobenius number$F(\boldsymbol{a})$of a lattice point$\boldsymbol{a}$in$\mathbb{R}^{d}$with positive coprime coordinates, is the largest integer which cannotbe expressed as a non-negative integer linear combination of the coordinates of$\boldsymbol{a}$. Marklof in [The asymptotic distribution of Frobenius numbers, Invent. Math.181(2010), 179–207] proved the existence of the limit distribution of the Frobenius numbers, when$\boldsymbol{a}$is taken to be random in an enlarging domain in$\mathbb{R}^{d}$. We will show that if the domain has piecewise smooth boundary, the error term for the convergence of the distribution function is at most a polynomial in the enlarging factor.

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