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

Harold G. Diamond, H. Halberstam, and William F. Galway
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Tracts in Mathematics 177
[Reviewed by
Allen Stenger
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This is a research monograph that develops a new sieve method. The "dimension" that appears in the title is also called the sifting density, and roughly speaking means the average number of residue classes sifted out for each prime p. For example, in the Eratosthenes sieve we eliminate all multiples of p, so we always eliminate the residue class 0 and the dimension is 1. For counting twin primes we seek integers n such that neither n nor n + 2 is divisible by p, so we eliminate the two classes 0 and –2 and the sieve has dimension 2. The dimension one sieves are much better understood than the higher-dimensional cases.

The central result, proved in Chapter 9, is an extension of the Jurkat-Richert Theorem to dimensions higher than 1. Sieve methods generate very complicated sums that need to be estimated to get a useful result, and it turns out that the sums can be approximated by the solutions to certain difference-differential equations (specifically differential delay equations), and the last half of the book is devoted to an analytic study of these equations. There is an appendix that considers the numerical solution of these equations using Mathematica, and the Mathematica package can be downloaded from the book's web site.

This is a well-crafted book, with clear writing. The authors break off every few pages and tell us where we are going and what the key upcoming result is. After each major result they work out an example of an important problem (such as the twin prime problem) to test how powerful the result is. Although the book starts at the beginning of sieve theory, it moves quickly into advanced territory, so it is not a book for beginners. Better choices to get started with sieves are Murty's Problems in Analytic Number Theory and Melvyn B. Nathanson's Additive Number Theory: The Classical Bases.


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.