Opera de Cribro

The book’s title is Latin and may be translated freely as “Sieve Theory: The Musical.” While the musical jokes and puns may not have you tapping your toes to the music of the primes, the book is a chatty and relatively painless way to get up to date on sieve theory. Our understanding of sieves has improved greatly over the past 20 or 30 years, in large part due to the efforts of this book’s two authors, and proofs that used to take many pages in an almost incomprehensible notation can now be done cleanly in one page. This book does a good job of keeping the notation under control.

A particular strength of the book is that it is loaded with applications. These include the field’s most famous results, such as Chen’s theorem on twin almost-primes, the Goldston-Pintz-Yildirim result on small gaps between consecutive primes, and the Friedlander-Iwaniec result that there are infinitely many primes of the form x² + y⁴.

The book has the flavor of a survey article, in that it tries to touch on all the most important subjects but is not complete or systematic. It is obviously much longer than a survey article, and unlike a survey it does include proofs for nearly everything discussed. The book includes many insights on where the various ideas came from and how they might be extended. The organization of the book makes it easy to skim, but the drawback is that (exacerbated by a skimpy index) it is often difficult to look up a particular result. It’s not an easy book to dip into. This is not a good book for beginners in sieve theory, because it is too concise and has no exercises. A good choice for beginners is Cojocaru and Murty’s An Introduction to Sieve Methods and Their Applications .

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.