Sieve theory has a rich and romantic history. The ancient question of whether there exist infinitely many twin primes (primes p such that p+2 is also prime), and Goldbach's conjecture that every even number can be written as the sum of two prime numbers, have been two of the problems that have inspired the development of the theory. This book provides a motivating introduction to sieve theory. Rather than focus on technical details which can obscure the beauty of the theory, the authors focus on examples and applications, developing the theory in parallel. The text can be used for a senior level undergraduate course or an introductory graduate course in analytic number theory.

### Contents

1. Some basic notions; 2. Some elementary sieves; 3. The normal order method; 4. The Turan sieve; 5. The sieve of Eratosthenes; 6. Brun's sieve; 7. Selberg's sieve; 8. The large sieve; 9. The Bombieri-Vinogradov theorem; 10. The lower bound sieve; 11. New directions in sieve theory; Bibliography.

### Reviews

"This book provides a sort of unifying theory of sieve methods, thus providing an indispensable resource for undergraduate and graduate students."
*MAA Reviews, Donald L. Vestal, Missouri Western State University*

"This book is directed towards graduate students or advanced undergraduates, and arises from a series of seminars and graduate lectures given over the last decade or so. The interested student should find that this book is a useful introduction to the subject that makes an effort to avoid those aspects of the subject which might be unduly heavy going. There is a useful set of exercises on each chapter that should keep such a student's mind suitably active."
*Mathematical Reviews*

"At only 224 pages, it is the shortest and simplest book on sieve methods that I have seen... beginners will appreciate the clear path laid out towards the modern theory. I enthusiastically recommend their book to any newcomer to the subject."
*Frank Thorne, University of South Carolina for the Bulletin of the American Mathematical Society*