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Probabilistic Number Theory II Central Limit Theorems

In this volume we study the value distribution of arithmetic functions, allowing unbounded renormalisations. The methods involve a synthesis of Probability and Number Theory; sums of independent infinitesimal random variables playing an important role. A central problem is to decide when an additive...

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Bibliographic Details
Main Author: Elliott, P.D.T.A.
Format: eBook
Language:English
Published: New York, NY Springer New York 1980, 1980
Edition:1st ed. 1980
Series:Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
Subjects:
Number Theory
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • Example of Rényi
  • Hardy—Ramanujan Estimate
  • Local Behaviour of Additive Functions Which Assume Values 0 and 1
  • Remarks and Examples
  • Connections with Hardy and Ramanujan Inequality
  • Uniform Local Upper Bound (Theorem (21.5))
  • Concluding Remarks
  • 22. The Distribution of the Quadratic Class Number
  • Statement of the Theorem
  • Approximation by Finite Euler Products
  • An Application of Duality
  • Construction of the Finite Probability Spaces
  • Approximation by Sums of Independent Random Variables
  • Concluding Remarks
  • 23 Problems
  • References (Roman)
  • References (Cyrillic)
  • Author Index
  • Volume II
  • 11. Unbounded Renormalisations: Preliminary Results
  • 12. The Erdös-Kac Theorem. Kubilius Models
  • Definition of Class H
  • Statement of Kubilius’ Main Theorem
  • Archetypal Application of a Kubilius Model
  • Analogue of the Feller—Lindeberg Condition
  • The Erdös-Kac Theorem
  • Turán’s Letter
  • Remarks upon Turan’s letter; LeVeque’s Conjecture
  • Erdös at Kac’ Lecture
  • Kac’ Letter
  • Remarks upon Kac’ Letter
  • Further Examples
  • Analogues on Shifted Primes
  • Example
  • Further Analogues on Shifted Primes, Application of Lévy’s Distance Function
  • Examples
  • Additive Functions on the Sequence N-p, p Prime
  • Barban’s Theorem on the Normal Order of f(p + 1)
  • Additive Functions on Polynomials
  • Additive Functions on Polynomials with Prime Arguments
  • Further Theorems and Examples
  • Quantitative Form of the Application of a Kubilius Model
  • Concluding Remarks
  • 13. The Weak Law of Large Numbers. I
  • Convergence to Normal Law
  • Convergence to Cauchy Law
  • Fractional Part of p ? 2, p Prime 13?
  • Construction of the Stable Laws
  • The Cauchy Law
  • Concluding Remarks
  • 17. The Limit Laws and the Renormalising Functions
  • Growth of?(x), (Theorem (17.1))
  • Class M Laws
  • Continuity of Limit Law (Theorem (17.2))
  • Laws of Class L are Absolutely Continuous (Lemma (17.11), Zolotarev)
  • Laws Which Cannot Occur
  • The Poisson Law
  • Further Continuity Properties
  • Conjectures
  • Conjectures (Summing Up)
  • 18. General Laws for Additive Functions. II: Logarithmic Renormalisation
  • Statement of the Main Theorems
  • Example of Erdös
  • Non-infinitely Divisible Law
  • Concluding Remarks
  • 19. Quantitative Mean-Value Theorems
  • Statement of the Main Results
  • Reduction to Application of Parseval’s Theorem (Lemma (19.5))
  • Upper Bounds for Dirichlet Series (Lemma (19.6))
  • The Prime Number Theorem
  • Axer’s Lemma (Lemma (19.8))
  • Primes in Arithmetic Progression; Character Sums
  • L-Series Estimates (Theorem (19.9))
  • The Position of the Elementary Proof of the Prime Number Theorem in the Theory of Arithmetic Functions
  • Hardy’s Copenhagen Remarks
  • Bohr’s Address at the International Mathematics Congress
  • Elementary Proof of Prime Number Theorem
  • Method of Delange
  • Method of Wirsing
  • Theorem of Wirsing
  • Historical Remark on the Application of Parseval’s Identity
  • Ingham’s Review
  • Concluding Remarks
  • 20. Rate of Convergence to the Normal Law
  • Theorem of Kubilius and Improvements (Theorem (20.1))
  • Examples
  • Additive Functions on Polynomials
  • Additive Functions on Polynomials with Prime Arguments
  • Examples
  • Conjugate Problem (Theorem (20.4))
  • Example
  • Improved Error Term for a Single Additive Function
  • Statement of the Main Theorem, (Theorem (20.5))
  • Examples
  • Concluding Remarks
  • 21. Local Theorems for Additive Functions
  • Existence of Densities
  • Theorem Concerning the Approximation of Additive Functions by Sums of Independent Random Variables
  • Essential Lemma (Lemma 13.2)
  • Concluding Remark
  • 14. The Weak Law of Large Numbers. II
  • Statement of the Main Results
  • The Approximate Functional Equation for ?(x)
  • of Haar Measures
  • of Dirichlet Series, Fourier Analysis on R
  • Study of the Integrals J
  • Approximate Differential Equation
  • A Compactness Lemma
  • Solution of the Differential Equation
  • Further Study of Dirichlet Series
  • The Decomposition of ?(x)
  • Proof of Theorem (14.1) (Necessity)
  • Proof of Theorem (14.1) (Sufficiency)
  • Proof of Theorem (14.2)
  • Concluding Remark
  • 15. A Problem of Hardy and Ramanujan
  • Theorems of Birch and Erdös
  • The Hardy—Ramanujan Problem. Statement of Theorem
  • Commentary on the Method of Turán
  • Examples
  • Concluding Remarks
  • 16. General Laws for Additive Functions. I: Including the Stable Laws
  • Statement of Isomorphism Theorem
  • Stable Laws

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