<p>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.</p>

A Higher-Dimensional Sieve Method: With Procedures for Computing Sieve Functions (Cambridge Tracts in Mathematics, Series Number 177)

Science & Math

Diamond, Harold G.

Halberstam, H.

Galway, William F.

Mathematics

Cambridge University Press

THIS volume is concerned with a substantial branch of number theory of which no connected account appears to exist; we describe the general nature of the constituent topics in the introduction. Although some excellent surveys dealing with limited aspects of the subject under con­ sideration have been published, the literature as a whole is far from easy to study. This is due in part to the extent of the literature; it is necessary to thread one's way through a maze of results, a complicated structure of inter-relationships, and many conflicting notations. In addition, however, not all the original papers are free from obscurities, and consequently some of these papers are difficult (a few even exceed­ ingly difficult) to master. We try to give a readable and coherent account of the subject, con­ taining a cross-section of the more interesting results. We felt that it would have been neither practicable nor desirable to attempt a compre­ hensive account; we treat each aspect of the subject from some special point of view, and select results accordingly. Needless to say, this approach entails the omission of many interesting and important results (quite apart from defects in the selection due to errors of judgement on our part). Those results selected for inclusion are, however, proved in complete detail and without the assumption of any prior knowledge on the part of the reader.

Sequences

Roth, K.F.

Springer

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