Probabilistic Number Theory I Mean-Value Theorems [Paperback]

Volume I.- About This Book.- 1. Necessary Results from Measure Theory.- Steinhaus Lemma.- Cauchys Functional Equation.- Slowly Oscillating Functions.- Halasz Lemma.- Fourier Analysis on the Line: Plancherels Theory.- The Theory of Probability.- Weak Convergence.- L?vys Metric.- Characteristic Functions.- Random Variables.- Concentration Functions.- Infinite Convolutions.- Kolmogorovs Inequality.- L?vys Continuity Criterion.- Purity of Type.- Wieners 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.- Selbergs Sieve Method; a Fundamental Lemma.- Upper Bound.- Lower Bound.- Distribution of Prime Numbers.- Dirichlet Series.- Euler Products.- Riemann Zeta Function.- WienerIkehara Tauberian Theorem.- HardyLittlewood Tauberian Theorem.- Quadratic Class Number, Dirichlets Identity.- Concluding Remarks.- 3. Finite Probability Spaces.- The Model of Kubilius.- Large Deviation Inequality.- A General Model.- Multiplicative Functions.- Concluding Remarks.- 4. The Tur?n-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 Erd?sWintner Theorem.- The Erd?sWintner Theorem.- Examples ?(n),?(n).- Limiting Distributions with Finite Mean and Variance.- The Function ?(n).- Modulus of Continuity, an Example of an Erd?s Proof.- Commentary on Erd?s Proof.- Concluding Remarks.- Alternative Proof of the Continuity of the Limit Law.- 6. Theorems of Delange, Wirsing, and Hal?sz.- Statement of the Main Theorems.- Application of ParsevallCo