A Higher-Dimensional Sieve Method: With Procedures for Computing Sieve Functions

by Harold G. Diamond, H. Halberstam, William F. Galway

Overview

As probability and combinatorics have penetrated the fabric of mathematical activity, sieve methods have become more versatile and sophisticated and in recent years have played a part in some of the most spectacular mathematical discoveries. Nearly a hundred years have passed since Viggo Brun invented his famous sieve, and the use of sieve methods is constantly evolving. Many arithmetical investigations encounter a combinatorial problem that requires a sieving argument, and this tract offers a modern and reliable guide in such situations. The theory of higher dimensional sieves is thoroughly explored, and examples are provided throughout. A Mathematica® software package for sieve-theoretical calculations is provided on the authors' website. To further benefit readers, the Appendix describes methods for computing sieve functions.

Product Details

ISBN-13:	9780521894876
Publisher:	Cambridge University Press
Publication date:	10/16/2008
Series:	Cambridge Tracts in Mathematics , #177
Pages:	290
Product dimensions:	6.20(w) x 9.00(h) x 1.00(d)

About the Author

Harold G. Diamond is Professor Emeritus in the Department of Mathematics at the University of Illinois at Urbana-Champaign.

Heini Halberstam is Professor Emeritus in the Department of Mathematics at the University of Illinois at Urbana-Champaign.

William F. Galway's research focuses on analytic and computational number theory. He is a member of the American Mathematical Society and of the Mathematical Association of America.

Table of Contents

List of tables; List of illustrations; Preface; Notation; Part I. Sieves: 1. Introduction; 2. Selberg's sieve method; 3. Combinatorial foundations; 4. The fundamental Lemma; 5. Selberg's sieve method (continued); 6. Combinatorial foundations (continued); 7. The case κ = 1: the linear sieve; 8. An application of the linear sieve; 9. A sieve method for κ > 1; 10. Some applications of Theorem 9.1; 11. A weighted sieve method; Part II. Proof of the Main Analytic Theorem: 12. Dramatis personae and preliminaries; 13. Strategy and a necessary condition; 14. Estimates of σκ (u) = jκ (u/2); 15. The pκ and qκ functions; 16. The zeros of Π−2 and Ξ; 17. The parameters σκ and βκ; 18. Properties of Fκ and fκ; Appendix 1. Methods for computing sieve functions; Bibliography; Index.

Editorial Reviews

"This is a well crafted book, with clear writing."
Allen Stenger, MAA Reviews

"It is to be recommended on the one hand for the serious student of the subject, and on the other for those who want a reference to the strongest available results for applications."
D.R. Heath-Brown, Mathematical Reviews

From the Publisher