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AuthorElliott, P. D. T. A. author
TitleProbabilistic Number Theory I [electronic resource] : Mean-Value Theorems / by P. D. T. A. Elliott
ImprintNew York, NY : Springer New York, 1979
Connect tohttp://dx.doi.org/10.1007/978-1-4612-9989-9
Descript 393 p. online resource

CONTENT

Volume I -- About This Book -- 1. Necessary Results from Measure Theory -- Steinhausโ{128}{153} Lemma -- Cauchyโ{128}{153}s Functional Equation -- Slowly Oscillating Functions -- Halaszโ{128}{153} Lemma -- Fourier Analysis on the Line: Plancherelโ{128}{153}s Theory -- The Theory of Probability -- Weak Convergence -- Lรฉvyโ{128}{153}s Metric -- Characteristic Functions -- Random Variables -- Concentration Functions -- Infinite Convolutions -- Kolmogorovโ{128}{153}s Inequality -- Lรฉvyโ{128}{153}s Continuity Criterion -- Purity of Type -- Wienerโ{128}{153}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 โ{128}{148} Stieltjes Transforms -- Distribution Functions (mod 1) -- Quantitative Fourier Inversion -- Berry-Esseen Theorem -- Concluding Remarks -- 2. Arithmetical Results, Dirichlet Series -- Selbergโ{128}{153}s Sieve Method; a Fundamental Lemma -- Upper Bound -- Lower Bound -- Distribution of Prime Numbers -- Dirichlet Series -- Euler Products -- Riemann Zeta Function -- Wienerโ{128}{148}Ikehara Tauberian Theorem -- Hardyโ{128}{148}Littlewood Tauberian Theorem -- Quadratic Class Number, Dirichletโ{128}{153}s 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 Erdรถsโ{128}{148}Wintner Theorem -- The Erdรถsโ{128}{148}Wintner 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{128}{153} 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 Parsevalโ{128}{153}s Formula -- Montgomeryโ{128}{153}s Lemma -- Product Representation of Dirichlet Series (Lemma 6.6) -- Quantitative form of Halรกszโ{128}{153} 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{128}{153} Conjecture -- The Nature of the Limit Law -- The Application of Schnirelmann Density -- Falsity of Erdรถsโ{128}{153} Conjecture -- Translation of Additive Functions (mod 1), Existence of Limiting Distribution -- Concluding Remarks -- 9. Mean Values of Multiplicative Functions, Halรกszโ{128}{153} Method -- Halรกszโ{128}{153} Main Theorem (Theorem (9.1)) -- Halรกszโ{128}{153} Lemma (Lemma (9.4)) -- Connections with the Large Sieve -- Halรกszโ{128}{153}s Second Lemma (Lemma (9.5)) -- Quantitative Form of Perronโ{128}{153}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 Turรกnโ{128}{148}Kubilius Inequality -- Study of Dirichlet Series -- Removal of the Condition p > p0 -- Application of a Method of Halรกsz -- Application of the Hardyโ{128}{148}Little wood Tauberian Theorem -- Application of a Theorem of Halรกsz -- Conclusion of Proof -- Concluding Remarks -- References (Roman) -- References (Cyrillic) -- Author Index xxm


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