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Sieves in Number Theory

  • Book
  • © 2001

Overview

Authors:
  1. George Greaves

    1. School of Mathematics, University of Wales, Cardiff Wales, UK

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  • Self-contained account of the "small" sieve method of V. Brun, A.Selberg, H. Iwaniec and others
  • Introduction and references for more specialised results in the modern literature
  • There is no direct competition
  • Includes supplementary material: sn.pub/extras

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Table of contents (8 chapters)

  1. Front Matter

    Pages I-XII
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  2. Introduction

    • George Greaves
    Pages 1-5

  3. The Structure of Sifting Arguments

    • George Greaves
    Pages 7-40

  4. Selberg’s Upper Bound Method

    • George Greaves
    Pages 41-70

  5. Combinatorial Methods

    • George Greaves
    Pages 71-101

  6. Rosser’s Sieve

    • George Greaves
    Pages 103-172

  7. The Sieve with Weights

    • George Greaves
    Pages 173-221

  8. The Remainder Term in the Linear Sieve

    • George Greaves
    Pages 223-258

  9. Lower Bound Sieves when κ > 1

    • George Greaves
    Pages 259-296

  10. Back Matter

    Pages 297-304
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Keywords

About this book

Slightly more than 25 years ago, the first text devoted entirely to sieve meth­ ods made its appearance, rapidly to become a standard source and reference in the subject. The book of H. Halberstam and H.-E. Richert had actually been conceived in the mid-1960's. The initial stimulus had been provided by the paper of W. B. Jurkat and Richert, which determined the sifting limit for the linear sieve, using a combination of the ,A2 method of A. Selberg with combinatorial ideas which were in themselves of great importance and in­ terest. One of the declared objectives in writing their book was to place on record the sharpest form of what they called Selberg sieve theory available at the time. At the same time combinatorial methods were not neglected, and Halber­ stam and Richert included an account of the purely combinatorial method of Brun, which they believed to be worthy of further examination. Necessar­ ily they included only a briefer mention of the development of these ideas due (independently) to J. B. Rosser and H. Iwaniec that became generally available around 1971. These combinatorial notions have played a central part in subsequent developments, most notably in the papers of Iwaniec, and a further account may be thought to be timely. There have also been some developments in the theory of the so-called sieve with weights, and an account of these will also be included in this book.

Reviews

From the reviews of the first edition:

"The author presents a self-contained account of the small sieve. … This well-written book will become my primary source for the small sieve … . I recommend it to everybody who is interested in the technically complicated theory on sieve methods." (R. Tijdeman, Nieuw Archief voor Wiskunde, Vol. 4 (3), 2003)

"The author’s choice of subjects provides a good background in the basic ideas of the sieve … . This text also supplies excellent background for some of the important unsolved problems of the subject. … In conclusion, the reviewer recommends this book strongly to students of sieve methods in the opening years of the twenty-first century. It will likely become one of the standard references on the subject." (Sidney W. Graham, Zentralblatt MATH, Vol. 1003 (03), 2003)

"The book being reviewed is an excellent survey on sieve methods. … The book is well written indeed, and most of the material can be described as self-contained. It can therefore be read by university graduates making their first acquaintance with the subject … ." (P. Shiu, The Mathematical Gazette, Vol. 86 (507), 2002)

Authors and Affiliations

  • School of Mathematics, University of Wales, Cardiff Wales, UK

    George Greaves

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