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Field name	Details
Dewey Class	512
Title	Probabilistic Number Theory I ([EBook]) : Mean-Value Theorems / by P. D. T. A. Elliott.
Author	Elliott, Peter D. T. A.
Other name(s)	SpringerLink (Online service)
Publication	New York, NY : Springer , 1979.
Physical Details	393 pages : online resource.
Series	Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 0072-7830 ; ; 239
ISBN	9781461299899
Contents note	Volume I -- About This Book -- 1. Necessary Results from Measure Theory -- Steinhaus’ Lemma -- Cauchy’s Functional Equation -- Slowly Oscillating Functions -- Halasz’ Lemma -- Fourier Analysis on the Line: Plancherel’s Theory -- The Theory of Probability -- Weak Convergence -- Lévy’s Metric -- Characteristic Functions -- Random Variables -- Concentration Functions -- Infinite Convolutions -- Kolmogorov’s Inequality -- Lévy’s Continuity Criterion -- Purity of Type -- Wiener’s Continuity Criterion -- Infinitely Divisible Laws -- Convergence of Infinitely Divisible Laws -- Limit Theorems for Sums of Independent Infinitesimal Random Variables -- Analytic Characteristic Functions -- The Method of Moments -- Mellin — Stieltjes Transforms -- Distribution Functions (mod 1) -- Quantitative Fourier Inversion -- Berry-Esseen Theorem -- Concluding Remarks -- 2. Arithmetical Results, Dirichlet Series -- Selberg’s Sieve Method; a Fundamental Lemma -- Upper Bound -- Lower Bound -- Distribution of Prime Numbers -- Dirichlet Series -- Euler Products -- Riemann Zeta Function -- Wiener—Ikehara Tauberian Theorem -- Hardy—Littlewood Tauberian Theorem -- Quadratic Class Number, Dirichlet’s Identity -- Concluding Remarks -- 3. Finite Probability Spaces -- The Model of Kubilius -- Large Deviation Inequality -- A General Model -- Multiplicative Functions -- Concluding Remarks -- 4. The Turán-Kubilius Inequality and Its Dual -- A Principle of Duality -- The Least Pair of Quadratic Non-Residues (mod p) -- Further Inequalities -- More on the Duality Principle -- The Large Sieve -- An Application of the Large Sieve -- Concluding Remarks -- 5. The Erdös—Wintner Theorem -- The Erdös—Wintner Theorem -- Examples ?(n),?(n) -- Limiting Distributions with Finite Mean and Variance -- The Function ?(n) -- Modulus of Continuity, an Example of an Erdös Proof -- Commentary on Erdös’ Proof -- Concluding Remarks -- Alternative Proof of the Continuity of the Limit Law -- 6. Theorems of Delange, Wirsing, and Halász -- Statement of the Main Theorems -- Application of Parseval’s Formula -- Montgomery’s Lemma -- Product Representation of Dirichlet Series (Lemma 6.6) -- Quantitative form of Halász’ Theorem for Mean-Value Zero -- Concluding Remarks -- 7. Translates of Additive and Multiplicative Functions -- Translates of Additive Functions -- Finitely Distributed Additive Functions -- The Surrealistic Continuity Theorem (Theorem 7.3) -- Additive Functions with Finite First and Second Means -- Distribution of Multiplicative Functions -- Criterion for Essential Vanishing -- Modified-weak Convergence -- Main Theorems for Multiplicative Functions -- Examples -- Concluding Remarks -- 8. Distribution of Additive Functions (mod 1) -- Existence of Limiting Distributions -- Erdös’ Conjecture -- The Nature of the Limit Law -- The Application of Schnirelmann Density -- Falsity of Erdös’ Conjecture -- Translation of Additive Functions (mod 1), Existence of Limiting Distribution -- Concluding Remarks -- 9. Mean Values of Multiplicative Functions, Halász’ Method -- Halász’ Main Theorem (Theorem (9.1)) -- Halász’ Lemma (Lemma (9.4)) -- Connections with the Large Sieve -- Halász’s Second Lemma (Lemma (9.5)) -- Quantitative Form of Perron’s Theorem (Lemma (9.6)) -- Proof of Theorem (9.1) -- Remarks -- 10. Multiplicative Functions with First and Second Means -- Statement of the Main Result (Theorem 10.1) -- Outline of the Argument -- Application of the Dual of the Turán—Kubilius Inequality -- Study of Dirichlet Series -- Removal of the Condition p > p0 -- Application of a Method of Halász -- Application of the Hardy—Little wood Tauberian Theorem -- Application of a Theorem of Halász -- Conclusion of Proof -- Concluding Remarks -- References (Roman) -- References (Cyrillic) -- Author Index xxm.
System details note	Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site	http://dx.doi.org/10.1007/978-1-4612-9989-9
Links to Related Works	Subject References: Algebra . Mathematics . Probablistic number theory . Authors: Elliott, Peter D. T. A. . Corporate Authors: SpringerLink (Online service) . Series: Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics . Classification: 512 .

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Probabilistic Number Theory I: Mean-Value Theorems

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