If you spend enough time in number theory circles (especially the analytic ones), you’re bound to run into a sieve or two. You learn about Brun’s sieve, Selberg’s sieve, or the “large sieve.” Details on these individual sieves can be found in scattered sources. This book provides a sort of unifying theory of sieve methods, thus providing an indispensable resource for undergraduate and graduate students.
The first chapter involves a nice presentation of some preliminaries, including Chebyshev’s Theorem, Bertrand’s postulate, and the Prime Number Theorem. The authors give both Ramanujan and Chebyshev’s proof of Chebyshev’s theorem, and then derive Bertrand’s postulate. The Prime Number Theorem is not proven, though it is stated (erroneously) as having been proved in 1895.
The authors present Gallagher’s larger sieve, the square sieve, Turán’s sieve, the sieve of Eratosthenes, Brun’s sieve, Selberg’s sieve, the large sieve, and the lower bound sieve. In addition, they discuss some of the connections between sieves and other mathematical objects: Dirichlet series, matrices (as developed by Elliott), and automorphic forms (as developed by Iwaniec). The last chapter (New directions in sieve theory) gives the material on linear algebra and automorphic forms. In future editions of this text, it would be fascinating to see this chapter expanded upon. Perhaps, in a few years, this material could end up requiring an additional volume.
As mentioned above, this text is a great resource. The authors also mention that it can be used as a textbook for a seminar or graduate course. The end of each of the eleven chapters contains some nice exercises, which lead the reader through further applications of the various methods presented in the text. And there are 75 references (“by no means exhaustive,” as the authors say) listed at the end of the text.
Donald L. Vestal is Associate Professor of Mathematics at Missouri Western State University. His interests include number theory, combinatorics, and a deep admiration for the crime-fighting efforts of the Aqua Teen Hunger Force. He can be reached at firstname.lastname@example.org
1. Some basic notions; 2. Some elementary sieves; 3. The normal order method; 4. The Turan sieve; 5. The sieve of Eratosthenes; 6. Brun’s sieve; 7. Selberg’s sieve; 8. The large sieve; 9. The Bombieri-Vinogradov theorem; 10. The lower bound sieve; 11. New directions in sieve theory; Bibliography.