Mean-value theorems

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.