Opera De Cribro (Colloquium Publications) (American Mathematical Society Colloquium Publications, 57)

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. —Roger Heath-Brown, University of Oxford, Fellow of Royal Society This is a comprehensive and up-to-date 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 developements 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. 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 Heath-Brown, University of Oxford, Fellow of Royal Society . This is a comprehensive and up-to-date 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. Chapter 1. Sieve Questions Chapter 2. Elementary Considerations On Arithmetic Functions Chapter 3. Bombieri's Sieve Chapter 4. Sieve Of Eratosthenes-legendre 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 Beta-sieve Chapter 12. The Linear Sieve Chapter 13. Applications To Linear Sequences Chapter 14. The Semi-linear 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. Almost-prime Sieve Appendix A. Mean Values Of Arithmetic Functions Appendix B. Differential-difference Equations John Friedlander, Henryk Iwaniec. Includes Bibliographical References (p. 519-523) And Index. In English. 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.-Roger Heath-Brown, University of Oxford, Fellow of Royal Society This