Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 43. Berlin: Springer (ISBN 3-540-41647-1/hbk). xii, 304 p. DM 189.00; £ 65.00; $ 99.00 (2001).
For students of sieves in the 1970s, the book Sieve Methods by H. Halberstam and H.-E. Richert (1974; Zbl 0298.10026) was the standard source and reference in the subject. It collected together the scattered results of the literature, and it standardized the notation and terminology. It also gave a comprehensive account of the known results on the representation of almost primes by polynomials and on corresponding results about the twin-prime conjecture and the Goldbach problem.
Twenty-five years later, Greaves has presented another text on sieve methods. Naturally, the subject has grown to the extent that is no longer possible to give a comprehensive treatment in one volume. The author has, however, made an appropriate choice of topics. The level of the text is appropriate for a graduate student with a one-semester course in analytic number theory. The author’s choice of subjects provides a good background in the basic ideas of the sieve, and at the same time, the text also presents some of the important developments that have taken place since the publication of Sieve Methods. This text also supplies excellent background for some of the important unsolved problems of the subject.
Chapter 1 (“The Structure of Sifting Arguments”) establishes basic notions and notation.
Chapter 2 (“Selberg’s Upper Bound Methods”) provides an introduction to the upper bound sieve of A. Selberg.
Chapter 3 (“Combinatorial Methods”) is an account of the combinatorial methods of Brun and Rosser. There are extremal examples, due to Selberg, which show that the Rosser sieve is optimal in dimension 1. Although the combinatorial methods preceded the method historically, the latter is simpler and leads more quickly to results of interest.
Chapter 4 (“Rosser’s Sieve”) is a thorough treatment of the analytic side of the Rosser sieve in arbitrary dimension. This chapter provides the novice reader with an example of how the analytic side of sieve methods connects with the arithmetic side. Rosser’s original treatment dates from the 1950s; it was contained in a long manuscript that was never published. The details were worked out by H. Iwaniec in a long paper published in 1980 [Acta Arith. 36, 171-202 (1980; Zbl 0435.10029)].
Chapter 5 (“The Sieve with Weights”) is the heart of the book; this is a nice account of the area where the author has made significant contributions. The chapter begins with an account of the Ankeny-Richert logarithmic weights. The author next introduces some modifications that lead to Buchstab’s and Laborde’s weights. This chapter culminates in the description of the “Greaves’ weighted sieve”, in which weights are introduced ab initio into the Rosser sieve. The presentation therefore emphasizes Greaves weighted sieve as a synthesis of the logarithmic weights and the Rosser sieve. Not surprisingly, the treatment requires a lot of technical details, but the author motivates the subject well. The outstanding open question for the weighted linear sieve is the “” conjecture. Selberg’s examples that were mentioned earlier also impose limitations on linear weighted sieves, and the “” conjecture states essentially that there are no further limitations on what can be attained. Greaves gives a very good discussion of this conjecture, the current state of knowledge, and the barriers to its resolution.
One of the most important themes in the theory of sieves in the last 25 years has been non-trivial treatment of remainder terms, and one of the most useful tools in doing this is Iwaniec’s bilinear form of the remainder term in the linear sieve. This is the topic of Chapter 6 (“The Remainder Term in the Linear Sieve”). Following earlier expositions by Motohashi and Halberstam and Richert, Greaves motivates Iwaniec’s construction by presenting it in terms of a condensation, or discretisation, of Rosser’s sieve.
Chapter 7 (“Lower Bound Sieves when ”) considers exposition of the Ankeny-Onishi sieve that results from combining the Selberg sieve with one Buchstab iteration. This gives better sifting limits than the Rosser sieve for dimension . Greaves also gives a brief account of the Selberg sieve, which gives even better sifting limits as tends to infinity. As the author notes, an important open question is to incorporate more of Selberg’s ideas to obtain improvements for smaller .
There are two important developments in sieve methods that are mentioned briefly in the book, but are not covered in detail. The first is the work of Diamond, Halberstam, and Richert on the limiting case of Buchstab’s iterative process combined with in dimension . The second is the “parity-breaking” work of Iwaniec and Friedlander, who established the infinitude of primes of the form , and D.R. Heath-Brown, who proved that there are infinitely many primes of the form Both of these topics deserve book-length treatment themselves, and Greaves quite appropriately left them out here.
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.