Volume II. - 11. Unbounded Renormalisations: Preliminary Results. - 12. The Erdös-Kac Theorem. Kubilius Models. - Definition of Class H. - Statement of Kubilius' Main Theorem. - Archetypal Application of a Kubilius Model. - Analogue of the FellerLindeberg Condition. - The Erdös-Kac Theorem. - Turán's Letter. - Remarks upon Turan's letter; LeVeque's Conjecture. - Erdös at Kac' Lecture. - Kac' Letter. - Remarks upon Kac' Letter. - Further Examples. - Analogues on Shifted Primes. - Example. - Further Analogues on Shifted Primes Application of Lévy's Distance Function. - Examples. - Additive Functions on the Sequence N-p p Prime. - Barban's Theorem on the Normal Order of f(p + 1). - Additive Functions on Polynomials. - Additive Functions on Polynomials with Prime Arguments. - Further Theorems and Examples. - Quantitative Form of the Application of a Kubilius Model. - Concluding Remarks. - 13. The Weak Law of Large Numbers. I. - Theorem Concerning the Approximation of Additive Functions by Sums of Independent Random Variables. - Essential Lemma (Lemma 13. 2). - Concluding Remark. - 14. The Weak Law of Large Numbers. II. - Statement of the Main Results. - The Approximate Functional Equation for ?(x). - of Haar Measures. - of Dirichlet Series Fourier Analysis on R. - Study of the Integrals J. - Approximate Differential Equation. - A Compactness Lemma. - Solution of the Differential Equation. - Further Study of Dirichlet Series. - The Decomposition of ?(x). - Proof of Theorem (14. 1) (Necessity). - Proof of Theorem (14. 1) (Sufficiency). - Proof of Theorem (14. 2). - Concluding Remark. - 15. A Problem of Hardy and Ramanujan. - Theorems of Birch and Erdös. - The HardyRamanujan Problem. Statement of Theorem. - Commentary on the Method of Turán. - Examples. - Concluding Remarks. - 16. General Laws for Additive Functions. I: Including the Stable Laws. - Statement of Isomorphism Theorem. - Stable Laws. - Convergence to Normal Law. - Convergence to Cauchy Law. - Fractional Part of p ? 2 p Prime 13?. - Construction of the Stable Laws. - The Cauchy Law. - Concluding Remarks. - 17. The Limit Laws and the Renormalising Functions. - Growth of?(x) (Theorem (17. 1)). - Class M Laws. - Continuity of Limit Law (Theorem (17. 2)). - Laws of Class L are Absolutely Continuous (Lemma (17. 11) Zolotarev). - Laws Which Cannot Occur. - The Poisson Law. - Further Continuity Properties. - Conjectures. - Conjectures (Summing Up). - 18. General Laws for Additive Functions. II: Logarithmic Renormalisation. - Statement of the Main Theorems. - Example of Erdös. - Non-infinitely Divisible Law. - Concluding Remarks. - 19. Quantitative Mean-Value Theorems. - Statement of the Main Results. - Reduction to Application of Parseval's Theorem (Lemma (19. 5)). - Upper Bounds for Dirichlet Series (Lemma (19. 6)). - The Prime Number Theorem. - Axer's Lemma (Lemma (19. 8)). - Primes in Arithmetic Progression; Character Sums. - L-Series Estimates (Theorem (19. 9)). - The Position of the Elementary Proof of the Prime Number Theorem in the Theory of Arithmetic Functions. - Hardy's Copenhagen Remarks. - Bohr's Address at the International Mathematics Congress. - Elementary Proof of Prime Number Theorem. - Method of Delange. - Method of Wirsing. - Theorem of Wirsing. - Historical Remark on the Application of Parseval's Identity. - Ingham's Review. - Concluding Remarks. - 20. Rate of Convergence to the Normal Law. - Theorem of Kubilius and Improvements (Theorem (20. 1)). - Examples. - Additive Functions on Polynomials. - Additive Functions on Polynomials with Prime Arguments. - Examples. - Conjugate Problem (Theorem (20. 4)). - Example. - Improved Error Term for a Single Additive Function. - Statement of the Main Theorem (Theorem (20. 5)). - Examples. - Concluding Remarks. - 21. Local Theorems for Additive Functions. - Existence of Densities. - Example of Rényi. - HardyRamanujan Estimate. - Local Behaviour of Additive Functions Which Assume Values 0 and 1. - Remarks and Examples. - Connections with Hardy and Ramanujan Inequality. - Uniform Local Upper Bound (Theorem (21. 5)). - Concluding Remarks. - 22. The Distribution of the Quadratic Class Number. - Statement of the Theorem. - Approximation by Finite Euler Products. - An Application of Duality. - Construction of the Finite Probability Spaces. - Approximation by Sums of Independent Random Variables. - Concluding Remarks. - 23 Problems. - References (Roman). - References (Cyrillic). - Author Index. Language: English