Despite its seemingly deterministic nature, the study of whole numbers, especially prime numbers, has many interactions with probability theory, the theory of random processes and events. This surprising connection was first discovered around 1920, but in recent years the links have become much deeper and better understood. Aimed at beginning graduate students, this textbook is the first to explain some of the most modern parts of the story. Such topics include the Chebychev bias, universality of the Riemann zeta function, exponential sums and the bewitching shapes known as Kloosterman paths. Emphasis is given throughout to probabilistic ideas in the arguments, not just the final statements, and the focus is on key examples over technicalities. The book develops probabilistic number theory from scratch, with short appendices summarizing the most important background results from number theory, analysis and probability, making it a readable and incisive introduction to this beautiful area of mathematics.
Author(s): Emmanuel Kowalski
Series: Cambridge Studies in Advanced Mathematics 192
Edition: 1
Publisher: Cambridge University Press
Year: 2021
Language: English
Commentary: Submitted Web Draft for Publication; Version of May 10, 2021; Available at https://people.math.ethz.ch/~kowalski/probabilistic-number-theory.html
Pages: 199
Tags: Number Theory; 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
