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.
An Introduction to Probabilistic Number Theory
1. Introduction; 2. Classical probabilistic number theory; 3. The distribution of values of the Riemann zeta function, I; 4. The distribution of values of the Riemann zeta function, II; 5. The Chebychev bias; 6. The shape of exponential sums; 7. Further topics; Appendix A. Analysis; Appendix B. Probability; Appendix C. Number theory; References; Index.