Probabilistic Number Theory I

Inhaltsverzeichnis

• 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.
• 3. Finite Probability Spaces.
• The Model of Kubilius.
• Large Deviation Inequality.
• A General Model.
• Multiplicative Functions.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• References (Roman).
• References (Cyrillic).
• Author Index xxm.