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Probabilistic number theory. II, Central limit theorems

Author: Peter D T A Elliott
Publisher: New York ; Berlin ; Heidelberg : Springer-Verlag, cop. 1980.
Series: Grundlehren der mathematischen Wissenschaften, 240.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:

In this volume we study the value distribution of arithmetic functions, allowing unbounded renormalisations. A central problem is to decide when an additive arithmetic function fin) admits a  Read more...

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Additional Physical Format: Probabilistic number theory. II, Central limit theorems / P.D.T.A. Elliott.
Paris : Université Paris Diderot, [201.]. (@Grundlehren der mathematischen Wissenschaften)
(ABES)186248695
Document Type: Book
All Authors / Contributors: Peter D T A Elliott
ISBN: 0387904387 9780387904382 3540904387 9783540904380 9781461299943 1461299942
OCLC Number: 490687967
Description: 1 vol. (XVIII-341-XXXIV p.) ; 25 cm.
Contents: Volume II.- 11. Unbounded Renormalisations: Preliminary Results.- 12. The Erdoes-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 Erdoes-Kac Theorem.- Turan's Letter.- Remarks upon Turan's letter; LeVeque's Conjecture.- Erdoes at Kac' Lecture.- Kac' Letter.- Remarks upon Kac' Letter.- Further Examples.- Analogues on Shifted Primes.- Example.- Further Analogues on Shifted Primes, Application of Levy'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.- 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 Erdoes.- The Hardy-Ramanujan Problem. Statement of Theorem.- Commentary on the Method of Turan.- Examples.- Concluding Remarks.- 16. General Laws for Additive Functions. I: Including the Stable Laws.- Statement of Isomorphism Theorem.- Stable Laws.- 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 Erdoes.- 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.- Example of Renyi.- 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.
Series Title: Grundlehren der mathematischen Wissenschaften, 240.
Responsibility: P.D.T.A. Elliott.

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