We conceived the idea of writing an introduction to applicable sieve theory during a brief encounter at Syracuse airport in the spring of 1966; we had
both been lecturing on sieves, and it seemed to us that the time for a
systematic exposition was long overdue. Our original plan was modest: to
to supply, in lecture notes form, the theoretical background for the method
of Jurkat-Richert [1] and to illustrate it by means of significant applica-
applications; also, to give a first connected account of the large sieve. As we
burrowed in the literature and came to realize its extent and wealth—the
notes we prepared on some two hundred papers in the summer of 1968 alone
would have filled a monograph—the inadequacy of our original conception
became increasingly clear. Not only did we find that we should have to
assimilate much classical material into a revised scheme but, as we
progressed, new results began to appear which affected it profoundly; so
much so that, viewing the matter in retrospect, if we had called a halt at any
earlier stage and published then, the book would have been much the poorer.
In the end we abandoned the chapters on the large sieve altogether—happily
the monographs of Montgomery [2] and Huxley [2] have appeared in the
meantime and provide admirable introductions to that subject and much else
beside—and concentrated on the "small" sieves of Brun and Selberg; even
sor critics should have no trouble in pointing to omissions, and we accept
any such criticisms in advance. Indeed, we have been at pains to point out
important extensions that time and space forced us to exclude, and we hope
only that what we have done here will make it easier for others to cover these
too in due course.