*(English)*Zbl 1207.11099

Let $\mathcal{P}$ be a finite set of primes. For a finite integer sequence $\mathcal{A}$ define

and for any $z\ge 1$,

The main purpose of the book is to give upper bounds (as well as lower bounds if possible) for the above two quantities. That goal is achieved via the so-called small sieve, a topic the authors have studied a lot (see e.g. the paper in collaboration with *H.-E. Richert* [“Combinatorial sieves of dimension exceeding one”, J. Number Theory 28, No. 3, 306–346 (1988; Zbl 0639.10031)] or the well-known [*H. Halberstam* and *H.-E. Richert*, Sieve methods. London: Academic Press, a subsidiary of Harcourt Brace Jovanovich, Publishers (1974; Zbl 0298.10026)]).

The first chapter of the book sets the most important notation and explains to what extent sieving techniques are well-suited for the study of various historical problems such as the twin prime conjecture.

The authors also define a parameter that plays a key role throughout the book: the dimension $\kappa $ of the sieving problem considered. Chapter 2 then gives a very clear description of Selberg’s sieve. Selberg’s Theorem is used several times in the rest of the book and is a key tool for the proof of the main results contained in the monograph. For instance the “fundamental Lemma” (cf Chapter 4), valid for any value of the parameter $\kappa $, is proved using Selberg’s sieve. Selberg’s sieve method is also the main topic of another chapter of the book: Chapter 5. In that Chapter, a refined version of Selberg’s Theorem is proved. Together with what the authors call ”the combinatorial foundations” (which have a lot to do with the linear sieve of Iwaniec and Rosser), it is definitely the technical heart of what is needed to obtain the results of the following chapters.

The rest of the book focuses on new upper bounds and lower bounds for $S(\mathcal{A},\mathcal{P},z)$. After considering the special case where $\kappa =1$ and discussing consequences for the twin prime conjecture, the authors state in Chapter 9 the main result of the book. It holds for any $\kappa $ such that $2\kappa $ is an integer. The upper bound (as well as the lower bound) obtained for $S(\mathcal{A},\mathcal{P},z)$ has the following shape:

where $y$ is a parameter such that $2\le y\le z$ and where ${g}_{\kappa}$ is not the same function for the upper bound and the lower bound.

Various applications of the main theorem are then given (e.g. a version of Mertens’ prime number formula for algebraic numbers), notably through a “weighted sieve” used to study the values taken by products of linear forms. The last part of the book deals with the study of the functions ${g}_{\kappa}$ appearing in the statement of the main result. Analytic properties of those functions are discussed. An appendix written by Galway describes computational techniques related to the sieving method the monograph deals with. Several tables are given so that the value of the sieving limit obtained via the new method can usefully be compared with previously known values (obtained e.g. by Iwaniec–Rosser).

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.