Volume I -- About This Book -- 1. Necessary Results from Measure Theory -- Steinhausโ Lemma -- Cauchyโs Functional Equation -- Slowly Oscillating Functions -- Halaszโ Lemma -- Fourier Analysis on the Line: Plancherelโs Theory -- The Theory of Probability -- Weak Convergence -- Lรฉvyโs Metric -- Characteristic Functions -- Random Variables -- Concentration Functions -- Infinite Convolutions -- Kolmogorovโs Inequality -- Lรฉvyโs Continuity Criterion -- Purity of Type -- Wienerโs 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 โ Stieltjes Transforms -- Distribution Functions (mod 1) -- Quantitative Fourier Inversion -- Berry-Esseen Theorem -- Concluding Remarks -- 2. Arithmetical Results, Dirichlet Series -- Selbergโs Sieve Method; a Fundamental Lemma -- Upper Bound -- Lower Bound -- Distribution of Prime Numbers -- Dirichlet Series -- Euler Products -- Riemann Zeta Function -- WienerโIkehara Tauberian Theorem -- HardyโLittlewood Tauberian Theorem -- Quadratic Class Number, Dirichletโs Identity -- Concluding Remarks -- 3. Finite Probability Spaces -- The Model of Kubilius -- Large Deviation Inequality -- A General Model -- Multiplicative Functions -- Concluding Remarks -- 4. The Turรกn-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 ErdรถsโWintner Theorem -- The ErdรถsโWintner Theorem -- Examples ?(n),?(n) -- Limiting Distributions with Finite Mean and Variance -- The Function ?(n) -- Modulus of Continuity, an Example of an Erdรถs Proof -- Commentary on Erdรถsโ Proof -- Concluding Remarks -- Alternative Proof of the Continuity of the Limit Law -- 6. Theorems of Delange, Wirsing, and Halรกsz -- Statement of the Main Theorems -- Application of Parsevalโs Formula -- Montgomeryโs Lemma -- Product Representation of Dirichlet Series (Lemma 6.6) -- Quantitative form of Halรกszโ 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 -- Erdรถsโ Conjecture -- The Nature of the Limit Law -- The Application of Schnirelmann Density -- Falsity of Erdรถsโ Conjecture -- Translation of Additive Functions (mod 1), Existence of Limiting Distribution -- Concluding Remarks -- 9. Mean Values of Multiplicative Functions, Halรกszโ Method -- Halรกszโ Main Theorem (Theorem (9.1)) -- Halรกszโ Lemma (Lemma (9.4)) -- Connections with the Large Sieve -- Halรกszโs Second Lemma (Lemma (9.5)) -- Quantitative Form of Perronโs 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 TurรกnโKubilius Inequality -- Study of Dirichlet Series -- Removal of the Condition p > p0 -- Application of a Method of Halรกsz -- Application of the HardyโLittle wood Tauberian Theorem -- Application of a Theorem of Halรกsz -- Conclusion of Proof -- Concluding Remarks -- References (Roman) -- References (Cyrillic) -- Author Index xxm

1. Mathematics
2. Algebra
3. Mathematics
4. Algebra