Slightly more than 25 years ago, the first text devoted entirely to
sieve meth- ods made its appearance, rapidly to become a standard source
and reference in the subject. The book of H. Halberstam and H.-E.
Richert had actually been conceived in the mid-1960's. The initial
stimulus had been provided by the paper of W. B. Jurkat and Richert,
which determined the sifting limit for the linear sieve, using a
combination of the, A2 method of A. Selberg with combinatorial ideas
which were in themselves of great importance and in- terest. One of the
declared objectives in writing their book was to place on record the
sharpest form of what they called Selberg sieve theory available at the
time. At the same time combinatorial methods were not neglected, and
Halber- stam and Richert included an account of the purely combinatorial
method of Brun, which they believed to be worthy of further examination.
Necessar- ily they included only a briefer mention of the development of
these ideas due (independently) to J. B. Rosser and H. Iwaniec that
became generally available around 1971. These combinatorial notions have
played a central part in subsequent developments, most notably in the
papers of Iwaniec, and a further account may be thought to be timely.
There have also been some developments in the theory of the so-called
sieve with weights, and an account of these will also be included in
this book.