Author | Elliott, P. D. T. A. author |
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Title | Probabilistic Number Theory II [electronic resource] : Central Limit Theorems / by P. D. T. A. Elliott |
Imprint | New York, NY : Springer US, 1980 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-9992-9 |
Descript | 375p. online resource |
Volume II -- 11. Unbounded Renormalisations: Preliminary Results -- 12. The Erdรถs-Kac Theorem. Kubilius Models -- Definition of Class H -- Statement of Kubiliusโ Main Theorem -- Archetypal Application of a Kubilius Model -- Analogue of the FellerโLindeberg Condition -- The Erdรถs-Kac Theorem -- Turรกnโs Letter -- Remarks upon Turanโs letter; LeVequeโs Conjecture -- Erdรถs at Kacโ Lecture -- Kacโ Letter -- Remarks upon Kacโ Letter -- Further Examples -- Analogues on Shifted Primes -- Example -- Further Analogues on Shifted Primes, Application of Lรฉvyโs Distance Function -- Examples -- Additive Functions on the Sequence N-p, p Prime -- Barbanโs 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 Erdรถs -- The HardyโRamanujan Problem. Statement of Theorem -- Commentary on the Method of Turรกn -- 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 Erdรถs -- Non-infinitely Divisible Law -- Concluding Remarks -- 19. Quantitative Mean-Value Theorems -- Statement of the Main Results -- Reduction to Application of Parsevalโs Theorem (Lemma (19.5)) -- Upper Bounds for Dirichlet Series (Lemma (19.6)) -- The Prime Number Theorem -- Axerโs 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 -- Hardyโs Copenhagen Remarks -- Bohrโs 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 Parsevalโs Identity -- Inghamโs 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 Rรฉnyi -- HardyโRamanujan 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