Arithmetic Randonn ́ ee An introduction to probabilistic number theory

Preface Prerequisites and notation Chapter 1. Introduction 1.1. Presentation 1.2. How does probability link with number theory really? 1.3. A prototype: integers in arithmetic progressions 1.4. Another prototype: the distribution of the Euler function 1.5. Generalizations 1.6. Outline of the book Chapter 2. Classical probabilistic number theory 2.1. Introduction 2.2. Distribution of arithmetic functions 2.3. The Erdos–Kac Theorem 2.4. Convergence without renormalization 2.5. Final remarks Chapter 3. The distribution of values of the Riemann zeta function, I 3.1. Introduction 3.2. The theorems of Bohr-Jessen and of Bagchi 3.3. The support of Bagchi's measure 3.4. Generalizations Chapter 4. The distribution of values of the Riemann zeta function, II 4.1. Introduction 4.2. Strategy of the proof of Selberg's theorem 4.3. Dirichlet polynomial approximation 4.4. Euler product approximation 4.5. Further topics Chapter 5. The Chebychev bias 5.1. Introduction 5.2. The Rubinstein–Sarnak distribution 5.3. Existence of the Rubinstein–Sarnak distribution 5.4. The Generalized Simplicity Hypothesis 5.5. Further results Chapter 6. The shape of exponential sums 6.1. Introduction 6.2. Proof of the distribution theorem 6.3. Applications 6.4. Generalizations Chapter 7. Further topics 7.1. Equidistribution modulo 1 7.2. Roots of polynomial congruences and the Chinese Remainder Theorem 7.3. Gaps between primes 7.4. Cohen-Lenstra heuristics 7.5. Ratner theory 7.6. And even more... Appendix A. Analysis A.1. Summation by parts A.2. The logarithm A.3. Mellin transform A.4. Dirichlet series A.5. Density of certain sets of holomorphic functions Appendix B. Probability B.1. The Riesz representation theorem B.2. Support of a measure B.3. Convergence in law B.4. Perturbation and convergence in law B.5. Convergence in law in a finite-dimensional vector space B.6. The Weyl criterion B.7. Gaussian random variables B.8. Subgaussian random variables B.9. Poisson random variables B.10. Random series B.11. Some probability in Banach spaces Appendix C. Number theory C.1. Multiplicative functions and Euler products C.2. Additive functions C.3. Primes and their distribution C.4. The Riemann zeta function C.5. Dirichlet L-functions C.6. Exponential sums Bibliography Index