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Nearly a hundred years have passed since Viggo Brun invented his famous sieve, and yet the use of sieve methods is still evolving. At one time it seemed that, as analytic tools improved, the use of sieves would decline, and only their role as an auxiliary device would survive. However, as probability and combinatorics have penetrated the fabric of mathematical activity, so have sieve methods become more versatile and sophisticated, especially in conjunction with other theories and methods, until, in recent years, they have played a part in some spectacular achievements that herald new directions in mathematical discovery.
An account of all the exciting and diverse applications of sieve ideas, past and present, has yet to be written. In this monograph our aim is modest and narrowly focused: we construct (in Chapter 9) a hybrid of the Selberg [Sel47] and Rosser-Iwaniec [Iwa80] sieve methods to deal with problems of sieve dimension (or density) that are integers or half integers. This theory achieves somewhat sharper estimates than either of its ancestors, the former as given by Ankeny and Onishi [AO65]. The sort of application we have in mind is to show that a given polynomial with integer coefficients (some obvious cases excluded) assumes at integers or at primes infinitely many almost-prime values, that is, values that have few prime factors relative to the degree of the polynomial.
Nearly a hundred years have passed since Viggo Brun invented his famous sieve, and the use of sieve methods is constantly evolving. 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. 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. These methods are generally applicable to the computation of other functions used in analytic number theory. The appendix also illustrates features of Mathematica® which aid in the computation of such functions.