Nearly a century after the discovery of Brun's sieve, we can look back and see how the subject has developed, and, to some extent, indicate how it may develop in the next 100 years.

One of the dominant themes of the twentieth century number theory has been the ‘modular connection’. In 1955, Yutaka Taniyama (1927–58) first hinted at a connection between elliptic curves and automorphic forms. The Langlands program has absorbed this theme and the connection is expected to hold in a wider context. At the heart of the Langlands program lies the ‘Rankin–Selberg method’, which signals an ‘orthogonality principle’ for automorphic representations on GL(n). This point of view has suggested one mode of generalizing the large sieve inequalities of analytic number theory.

In a series of remarkable papers, H. Iwaniec and his school have developed the ‘modular connection’ and the cognate ‘spectral connection’ as it applies to GL(2) analogues of the large sieve inequality (see, for example,).

In this chapter, we give a brief overview of the work of Duke and Kowalski that suggests a future direction for the large sieve method. No doubt, there will be other directions of development, but the authors do not have a crystal ball to perceive them.

A duality principle

The large sieve inequality can be reduced to a statement in linear algebra, which in turn can be proven using matrix theory.

Viggo Brun (1885–1978) introduced the sieve that now bears his name in 1915 in the paper. It seems that Jean Merlin had made the first serious attempt to go beyond Eratosthenes. Unfortunately, he was killed in World War I (see) and only two of his manuscripts have survived, namely. The latter was prepared for publication by Jacques Hadamard (1865–1963) and published posthumously. Clearly, Brun read Merlin's papers very carefully and was inspired by them. Perhaps he was the only one to have done so. No doubt, this led to his 1915 paper on the subject and later these results were developed into a sophisticated sieve.

In his fundamental work, Brun proved that there are infinitely many integers n such that n and n + 2 have at most nine prime factors. He also showed that all sufficiently large even integers are the sum of two integers, each having at most nine prime factors. These represent tremendous advances towards the twin prime conjecture and the Goldbach conjecture. As a consequence of this work, he deduced that the sum of the reciprocals of the sequence of twin primes converges (see Corollary 5.4.5).

Brun's original papers on the subject were largely ignored. One story reports that Edmund Landau (1877–1938) had not looked at them for eight years, even though they were on his desk. Part of the difficulty lay in the unwieldy notation that Brun had used. Now, 80 years later, the ideas look simple enough and the notation has been streamlined.

It is now nearly 100 years since the birth of modern sieve theory. The theory has had a remarkable development and has emerged as a powerful tool, not only in number theory, but in other branches of mathematics, as well. Until 20 years ago, three sieve methods, namely Brun's sieve, Selberg's sieve and the large sieve of Linnik, could be distinguished as the major pillars of the theory. But after the fundamental work of Deshouillers and Iwaniec in the 1980's, the theory has been linked to the theory of automorphic forms and the fusion is making significant advances in the field.

This monograph is the outgrowth of seminars and graduate courses given by us during the period 1995–2004 at McGill and Queen's Universities in Canada, and Princeton University in the US. Its singular purpose is to acquaint graduate students to the difficult, but extremely beautiful area, and enable them to apply these methods in their research. Hence we do not develop the detailed theory of each sieve method. Rather, we choose the most expedient route to introduce it and quickly indicate various applications. The reader may find in the literature more detailed and encyclopedic accounts of the theory (many of these are listed in the references). Our purpose here is didactic and we hope that many will find the treatment elegant and enjoyable.