The Distribution of Prime Numbers (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 203)

Prime numbers have fascinated mathematicians since the time of Euclid. This book presents some of our best tools to capture the properties of these fundamental objects, beginning with the most basic notions of asymptotic estimates and arriving at the forefront of mathematical research. Detailed proofs of the recent spectacular advances on small and large gaps between primes are made accessible for the first time in textbook form. Some other highlights include an introduction to probabilistic methods, a detailed study of sieves, and elements of the theory of pretentious multiplicative functions leading to a proof of Linnik's theorem. Throughout, the emphasis has been placed on explaining the main ideas rather than the most general results available. As a result, several methods are presented in terms of concrete examples that simplify technical details, and theorems are stated in a form that facilitates the understanding of their proof at the cost of sacrificing some generality. Each chapter concludes with numerous exercises of various levels of difficulty aimed to exemplify the material, as well as to expose the readers to more advanced topics and point them to further reading sources. Dedication Contents Preface Notation And then there were infinitely many Part 1. First principles 1. Asymptotic estimates 2. Combinatorial ways to count primes 3. The Dirichlet convolution 4. Dirichlet series Part 2. Methods of complex and harmonic analysis 5. An explicit formula for counting primes 6. The Riemann zeta function 7. The Perron inversion formula 8. The Prime Number Theorem 9. Dirichlet characters 10. Fourier analysis on finite abelian groups 11. Dirichlet L-functions 12. The Prime Number Theorem for arithmetic progressions Part 3. Multiplicative functions and the anatomy of integers 13. Primes and multiplicative functions 14. Evolution of sums of multiplicative functions 15. The distribution of multiplicative functions 16. Large deviations Part 4. Sieve methods 17. Twin primes 18. The axioms of sieve theory 19. The Fundamental Lemma of Sieve Theory 20. Applications of sieve methods 21. Selberg’s sieve 22. Sieving for zero-free regions Part 5. Bilinear methods 23. Vinogradov’s method 24. Ternary arithmetic progressions 25. Bilinear forms and the large sieve 26. The Bombieri-Vinogradov theorem 27. The least prime in an arithmetic progression Part 6. Local aspects of the distribution of primes 28. Small gaps between primes 29. Large gaps between primes 30. Irregularities in the distribution of primes Appendices Appendix A. The Riemann-Stieltjes integral Appendix B. The Fourier and the Mellin transforms Appendix C. The method of moments Bibliography Index

Prime numbers have fascinated mathematicians since the time of Euclid. This book presents some of our best tools to capture the properties of these fundamental objects, beginning with the most basic notions of asymptotic estimates and arriving at the forefront of mathematical research. Detailed proofs of the recent spectacular advances on small and large gaps between primes are made accessible for the first time in textbook form. Some other highlights include an introduction to probabilistic methods, a detailed study of sieves, and elements of the theory of pretentious multiplicative functions leading to a proof of Linnik's theorem.Throughout, the emphasis has been placed on explaining the main ideas rather than the most general results available. As a result, several methods are presented in terms of concrete examples that simplify technical details, and theorems are stated in a form that facilitates the understanding of their proof at the cost of sacrificing some generality. Each chapter concludes with numerous exercises of various levels of difficulty aimed to exemplify the material, as well as to expose the readers to more advanced topics and point them to further reading sources.