Mean-value theorems
Author(s)
Bibliographic Information
Mean-value theorems
(Die Grundlehren der mathematischen Wissenschaften, 239 . Probabilistic number theory / P. D. T. A. Elliott ; 1)
Springer-Verlag, c1979
- : us
- : gw
Available at 96 libraries
  Aomori
  Iwate
  Miyagi
  Akita
  Yamagata
  Fukushima
  Ibaraki
  Tochigi
  Gunma
  Saitama
  Chiba
  Tokyo
  Kanagawa
  Niigata
  Toyama
  Ishikawa
  Fukui
  Yamanashi
  Nagano
  Gifu
  Shizuoka
  Aichi
  Mie
  Shiga
  Kyoto
  Osaka
  Hyogo
  Nara
  Wakayama
  Tottori
  Shimane
  Okayama
  Hiroshima
  Yamaguchi
  Tokushima
  Kagawa
  Ehime
  Kochi
  Fukuoka
  Saga
  Nagasaki
  Kumamoto
  Oita
  Miyazaki
  Kagoshima
  Okinawa
  Korea
  China
  Thailand
  United Kingdom
  Germany
  Switzerland
  France
  Belgium
  Netherlands
  Sweden
  Norway
  United States of America
Note
Includes bibliographies and indexes
Description and Table of Contents
Table of Contents
- 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.- Levy's Metric.- Characteristic Functions.- Random Variables.- Concentration Functions.- Infinite Convolutions.- Kolmogorov's Inequality.- Levy'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 Turan-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 Erdos-Wintner Theorem.- The Erdos-Wintner Theorem.- Examples ?(n),?(n).- Limiting Distributions with Finite Mean and Variance.- The Function ?(n).- Modulus of Continuity, an Example of an Erdos Proof.- Commentary on Erdos' Proof.- Concluding Remarks.- Alternative Proof of the Continuity of the Limit Law.- 6. Theorems of Delange, Wirsing, and Halasz.- Statement of the Main Theorems.- Application of Parseval's Formula.- Montgomery's Lemma.- Product Representation of Dirichlet Series (Lemma 6.6).- Quantitative form of Halasz' 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.- Erdos' Conjecture.- The Nature of the Limit Law.- The Application of Schnirelmann Density.- Falsity of Erdos' Conjecture.- Translation of Additive Functions (mod 1), Existence of Limiting Distribution.- Concluding Remarks.- 9. Mean Values of Multiplicative Functions, Halasz' Method.- Halasz' Main Theorem (Theorem (9.1)).- Halasz' Lemma (Lemma (9.4)).- Connections with the Large Sieve.- Halasz'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 Turan-Kubilius Inequality.- Study of Dirichlet Series.- Removal of the Condition p > p0.- Application of a Method of Halasz.- Application of the Hardy-Little wood Tauberian Theorem.- Application of a Theorem of Halasz.- Conclusion of Proof.- Concluding Remarks.- References (Roman).- References (Cyrillic).- Author Index xxm.
by "Nielsen BookData"