Sieve Methods

Sieve methods are an important tool of analytic number theory. Sieve Methods was the first major text on this method, which dates back to Eratothenees’ sieve and has continuously developed with new contributions in modern times, such as Brun’s sieve, Rosser’s sieve, linear sieves and weighted sieves.

Before further discussing the book proper, I indicate which audiences could benefit from this book. The following comments apply to any upper-level undergraduate course, 1st year graduate course, or advanced graduate course.

1) Reference: I would definitely have this book on reserve in the library as a reference text for any course on number theory.

2) General Number-Theory Course Textbook: If an instructor is teaching a general number-theory course, they could use multiple texts because number theory is a discipline with several well developed branches. So a typical one or two semester number theory course might devote 2–3 weeks each to basics, analytic methods (involving calculus), algebraic number-field techniques, discrete-mathematical methods, and sieve methods. Such a course might naturally have multiple texts, one for each module of the course. In such a case the first few chapters of Sieve Methods, which are very well written and easy to read, would fit in nicely.

However, alternative approaches to a general number theory course with several modules are also possible; one can also find good single texts which use this multiple module approach. An Invitation to Number Theory by Steven J. Miller and Ramin Takloo-Bighash, for example, provides in a single text modules on basics; transcendence, continued fractions and approximation; probabilistic methods; circle methods, L-functions. A drawback for using Sieve Methods as a course text is the total lack of problems.

3) A Course in Sieve Theory: I would not use Sieve Methods as the class text for a course on sieve theory. Although it is the classic text, well written, comprehensive and covers many topics, the field has grown vigorously and Sieve Methods is now outdated.

There are a several good books which would bring students to the current frontiers of the field, written by experts who have helped create the modern approach. Perhaps the best current book is Opera de Cribro, by Henryk Iwaniec and John Friedlander. That being said, Sieve Methods could still have a use in such a course as in many topics it would make for easier reading.

Perhaps an analogy would help communicate this point: you don’t necessarily use the classic text that introduced a field as a course textbook: One would not use Euclid’s Elements as a text for a geometry course nor Boole’s The Laws of Thought as the text for a logic course. The point here is that after the classic texts were written, more readable notation, simpler methods, and more unifying proofs were developed.

4) Researchers: Anyone doing research in sieve methods should definitely have this book. Because the text was the first comprehensive book on sieve methods, it contains a wealth of details and minutiae that can easily be overlooked in more current texts. A skillful researcher can very often revive an obscure method which is currently not being used.

Sieve Methods book was first published in 1974 and recently reprinted. It was the first major text on the subject. As a result, the authors faced the problem of developing a standard unified notation. They did this admirably and the book is very readable. The book included many previously unpublished methods and results, and had almost no prerequisites beyond knowledge of the real line and associated notations. For example, reference to characters and countour integration is left to the last chapter. This makes the book accessible to upper level undergraduates.

The book covers and unifies many interesting topics including Eratosthenes’ sieve, Brun’s sieve, Rosser’s sieve, linear sieves and weighted sieves. It includes many important theorems: Selberg’s upper bound, the Brun-Titschmarsh inequality, Chen’s theorem, etc. It explains many applications, such as the Goldback conjecture and its generalizations, twin primes and k-tuplets of primes, the Titschmarsh divisor problem, prime representation by polynomials, almost-primes and their representation etc.

Although the book has no exercises, it has a wealth of references. Close to 400 papers are referenced and classified under six general categories: theoretical contributions, surveys, methodological developments, applications, extensions to algebraic domains, variants, and aids. The bibliography and references are a very important contribution to the field and are relevant even today to researchers in the field.

Russell Jay Hendel (RHendel@Towson.Edu) holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.