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A Higher-Dimensional Sieve Method

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


Part of Cambridge Tracts in Mathematics

  • Date Published: October 2008
  • availability: In stock
  • format: Hardback
  • isbn: 9780521894876

£ 95.00

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About the Authors
  • 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.

    • Fully explains the theory of higher dimensional sieves using many examples
    • Modern and reliable guide for researchers needing to solve combinatorial problems with sieving arguments
    • Computational methods are explained in detail in an appendix and on the accompanying website
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    Reviews & endorsements

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

    'This is a well-crafted book, with clear writing.' MAA Reviews

    '… a very good source of details of techniques used in sieve methods and their applications, and therefore it can be warmly recommended for everyone interested in sieve methods.' EMS Newsletter

    'Overall, this monograph is quite technical and is mostly aimed at people with a reasonable background in sieving theory. However, it is an excellent reference for anyone searching for the most up-to-date results in the theory.' Zentralblatt MATH

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    Product details

    • Date Published: October 2008
    • format: Hardback
    • isbn: 9780521894876
    • length: 290 pages
    • dimensions: 235 x 155 x 20 mm
    • weight: 0.52kg
    • contains: 5 b/w illus. 15 tables
    • availability: In stock
  • Table of Contents

    List of tables
    List of illustrations
    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

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    A Higher-Dimensional Sieve Method

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

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  • Authors

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

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

    William F. Galway
    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.

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