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.