Archimedes has been quoted as saying, "Give me a fulcrum, a lever that is long enough, and a place to stand, and I will move the earth." After reading the book under review the reader could adapt these words to claim, "Give me sufficiently good Type I and Type II information and I will prove the twin prime conjecture, the Goldbach conjecture, and so on." This is how Harman concludes his book (page 335).
He begins the book by introducing the sieve of Eratosthenes, the more precise one of Eratosthenes-Legendre, and some others. The book has 14 chapters. With the exception of the first and last, each focuses on a method that can be used to attack four very important problems in number theory: Diophantine approximation, primes in short intervals, primes in arithmetic progressions, and primes represented by additive forms. However, the aim is not to study each of these problems separately; rather, as the title indicates, the book deals with methods that can be used for many problems about primes. The discussion is based on famous and important work of the author, John B. Friedlander, Roger Heath-Brown, Henryk Iwaniec, Robert C. Vaughan, and others.
All the results in the book are quite beautiful and strong, but I believe that the most impressive parts are those that consider the problem of primes in short intervals, proving some very critical results using various sieves. The author shows the power of sieve methods for attacking such famous problems as Goldbach cojecture.
The book, is quite technical and hard for an amateur reader. It can be used as a textbook (for sufficiently advanced students); the author provides exercises at the end of some chapters and an appendix containing required fundamental topics. But I think that the book is most useful for researchers in analytic number theory who would like to see the most recent methods, results and challenging problems in the area.
In summary, this is a serious book on sieve methods which contains a collection of very important results by a number of giants of prime number theory.
Mehdi Hassani is a "co-tutelle" Ph.D. student in Mathematics in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, and the Université de Bordeaux I, under supervision of the professors M.M. Shahshahani and J-M. Deshouillers.
Chapter 1. Introduction 1
Chapter 2. The Vaughan Identity 25
Chapter 3. The Alternative Sieve 47
Chapter 4. The Rosser-Iwaniec Sieve 65
Chapter 5. Developing the Alternative Sieve 83
Chapter 6. An Upper-Bound Sieve 103
Chapter 7. Primes in Short Intervals 119
Chapter 8. The Brun-Titchmarsh Theorem on Average 157
Chapter 9. Primes in Almost All Intervals 189
Chapter 10. Combination with the Vector Sieve 201
Chapter 11. Generalizing to Algebraic Number Fields 231
Chapter 12. Variations on Gaussian Primes 265
Chapter 13. Primes of the Form x3 + 2y3 303
Chapter 14. Epilogue 335