Hardcover ISBN:  9780821849705 
Product Code:  COLL/57 
List Price:  $103.00 
MAA Member Price:  $92.70 
AMS Member Price:  $82.40 
Electronic ISBN:  9781470417666 
Product Code:  COLL/57.E 
List Price:  $103.00 
MAA Member Price:  $92.70 
AMS Member Price:  $82.40 

Book DetailsColloquium PublicationsVolume: 57; 2010; 527 ppMSC: Primary 11;
This monograph represents the state of the art both in respect of coverage of the general methods and in respect of the actual applications to interesting problems.
A unique feature of this monograph is how the authors take great pains to explain the fundamental ideas behind the proofs and to show how to approach a question in a correct fashion. So, this book is not just another monograph useful for consultation; rather, it is a teaching instrument of great value both for the specialist and the beginner in the field.
The authors must be congratulated for this exceptional monograph, the first of its kind for depth of content as well as for the effort made to explain the ‘why’ and not limiting themselves to the ‘how to’. This is a true masterpiece that will prove to be indispensable to the serious researcher for many years to come.—Enrico Bombieri, Institute for Advanced Study
This is a truly comprehensive account of sieves and their applications, by two of the world's greatest authorities. Beginners will find a thorough introduction to the subject, with plenty of helpful motivation. The more practised reader will appreciate the authors' insights into some of the more mysterious parts of the theory, as well as the wealth of new examples. No analytic number theorist should be without this volume, but it will not have a place on my bookshelves—it will be permanently on my desk!
—Roger HeathBrown, University of Oxford,Fellow of Royal Society
This is a comprehensive and uptodate treatment of sieve methods. The theory of the sieve is developed thoroughly with complete and accessible proofs of the basic theorems. Included is a wide range of applications, both to traditional questions such as those concerning primes, and to areas previously unexplored by sieve methods, such as elliptic curves, points on cubic surfaces and quantum ergodicity. New proofs are given also of some of the central theorems of analytic number theory; these proofs emphasize and take advantage of the applicability of sieve ideas.
The book contains numerous comments which provide the reader with insight into the workings of the subject, both as to what the sieve can do and what it cannot do. The authors reveal recent developments by which the parity barrier can be breached, exposing golden nuggets of the subject, previously inaccessible. The variety in the topics covered and in the levels of difficulty encountered makes this a work of value to novices and experts alike, both as an educational tool and a basic reference.ReadershipGraduate students and research mathematicians interested in number theory.

Table of Contents

Chapters

Chapter 1. Sieve questions

Chapter 2. Elementary considerations on arithmetic functions

Chapter 3. Bombieri’s sieve

Chapter 4. Sieve of EratosthenesLegendre

Chapter 5. Sieve principles and terminology

Chapter 6. Brun’s sieve—The big bang

Chapter 7. Selberg’s sieve—Kvadrater er positive

Chapter 8. Sieving by many residue classes

Chapter 9. The large sieve

Chapter 10. Molecular structure of sieve weights

Chapter 11. The betasieve

Chapter 12. The linear sieve

Chapter 13. Applications to linear sequences

Chapter 14. The semilinear sieve

Chapter 15. Applications—Choice but not prime

Chapter 16. Asymptotic sieve and the parity principle

Chapter 17. Combinatorial identities

Chapter 18. Asymptotic sieve for primes

Chapter 19. Equidistribution of quadratic roots

Chapter 20. Marching over Gaussian primes

Chapter 21. Primes represented by polynomials

Chapter 22. Level of distribution of arithmetic sequences

Chapter 23. Primes in short intervals

Chapter 24. The least prime in an arithmetic progression

Chapter 25. Almostprime sieve

Appendix A. Mean values of arithmetic functions

Appendix B. Differentialdifference equations


Additional Material

Reviews

Our understanding of sieves has improved greatly over the past 20 or 30 years, in large part due to the efforts of this book's authors, and proofs that use to take many pages in an almost incomprehensible notation can now be done cleanly in one page. This book does a good job of keeping the notation under control.
MAA Reviews 
Written by the two leading authorities on the subject, it contains a wealth of insights as well as a string of new results. ... The topics covered here are remarkably wideranging, as are the connections which the authors make. Indeed this book is not just a volume on sieves, but rather a treatise on analytic number theory more generally. As such it is recommended to everyone with an interest in the subject.
Mathematical Reviews


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This monograph represents the state of the art both in respect of coverage of the general methods and in respect of the actual applications to interesting problems.
A unique feature of this monograph is how the authors take great pains to explain the fundamental ideas behind the proofs and to show how to approach a question in a correct fashion. So, this book is not just another monograph useful for consultation; rather, it is a teaching instrument of great value both for the specialist and the beginner in the field.
The authors must be congratulated for this exceptional monograph, the first of its kind for depth of content as well as for the effort made to explain the ‘why’ and not limiting themselves to the ‘how to’. This is a true masterpiece that will prove to be indispensable to the serious researcher for many years to come.
—Enrico Bombieri, Institute for Advanced Study
This is a truly comprehensive account of sieves and their applications, by two of the world's greatest authorities. Beginners will find a thorough introduction to the subject, with plenty of helpful motivation. The more practised reader will appreciate the authors' insights into some of the more mysterious parts of the theory, as well as the wealth of new examples. No analytic number theorist should be without this volume, but it will not have a place on my bookshelves—it will be permanently on my desk!
—Roger HeathBrown, University of Oxford,Fellow of Royal Society
This is a comprehensive and uptodate treatment of sieve methods. The theory of the sieve is developed thoroughly with complete and accessible proofs of the basic theorems. Included is a wide range of applications, both to traditional questions such as those concerning primes, and to areas previously unexplored by sieve methods, such as elliptic curves, points on cubic surfaces and quantum ergodicity. New proofs are given also of some of the central theorems of analytic number theory; these proofs emphasize and take advantage of the applicability of sieve ideas.
The book contains numerous comments which provide the reader with insight into the workings of the subject, both as to what the sieve can do and what it cannot do. The authors reveal recent developments by which the parity barrier can be breached, exposing golden nuggets of the subject, previously inaccessible. The variety in the topics covered and in the levels of difficulty encountered makes this a work of value to novices and experts alike, both as an educational tool and a basic reference.
Graduate students and research mathematicians interested in number theory.

Chapters

Chapter 1. Sieve questions

Chapter 2. Elementary considerations on arithmetic functions

Chapter 3. Bombieri’s sieve

Chapter 4. Sieve of EratosthenesLegendre

Chapter 5. Sieve principles and terminology

Chapter 6. Brun’s sieve—The big bang

Chapter 7. Selberg’s sieve—Kvadrater er positive

Chapter 8. Sieving by many residue classes

Chapter 9. The large sieve

Chapter 10. Molecular structure of sieve weights

Chapter 11. The betasieve

Chapter 12. The linear sieve

Chapter 13. Applications to linear sequences

Chapter 14. The semilinear sieve

Chapter 15. Applications—Choice but not prime

Chapter 16. Asymptotic sieve and the parity principle

Chapter 17. Combinatorial identities

Chapter 18. Asymptotic sieve for primes

Chapter 19. Equidistribution of quadratic roots

Chapter 20. Marching over Gaussian primes

Chapter 21. Primes represented by polynomials

Chapter 22. Level of distribution of arithmetic sequences

Chapter 23. Primes in short intervals

Chapter 24. The least prime in an arithmetic progression

Chapter 25. Almostprime sieve

Appendix A. Mean values of arithmetic functions

Appendix B. Differentialdifference equations

Our understanding of sieves has improved greatly over the past 20 or 30 years, in large part due to the efforts of this book's authors, and proofs that use to take many pages in an almost incomprehensible notation can now be done cleanly in one page. This book does a good job of keeping the notation under control.
MAA Reviews 
Written by the two leading authorities on the subject, it contains a wealth of insights as well as a string of new results. ... The topics covered here are remarkably wideranging, as are the connections which the authors make. Indeed this book is not just a volume on sieves, but rather a treatise on analytic number theory more generally. As such it is recommended to everyone with an interest in the subject.
Mathematical Reviews