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A Higher-Dimensional Sieve Method: With Procedures for Computing Sieve Functions (Cambridge Tracts in Mathematics)

A Higher-Dimensional Sieve Method: With Procedures for Computing Sieve Functions (Cambridge Tracts in Mathematics)

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Harold G. Diamond, H. Halberstam, William F. Galway
Cambridge University Press, 10/16/2008
EAN 9780521894876, ISBN10: 0521894875

Hardcover, 290 pages, 23.5 x 15.5 x 2 cm
Language: English

Nearly a hundred years have passed since Viggo Brun invented his famous sieve, and the use of sieve methods is constantly evolving. 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. 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. These methods are generally applicable to the computation of other functions used in analytic number theory. The appendix also illustrates features of Mathematica® which aid in the computation of such functions.

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.

'... definitely of great interest to the intermediate-advanced sieve theorist.' Internationale Mathematische Nachrichten