Opera de cribro

Includes index

Includes bibliography (p. 519-524) references.

1. Sieve questions
2. Elementary considerations on arithmetic functions
3. Bombieri’s sieve
4. Sieve of Eratosthenes-Legendre
5. Sieve principles and terminology
6. Brun’s sieve—The big bang
7. Selberg’s sieve—Kvadrater er positive
8. Sieving by many residue classes
9. The large sieve
10. Molecular structure of sieve weights
11. The beta-sieve
12. The linear sieve
13. Applications to linear sequences
14. The semi-linear sieve
15. Applications—Choice but not prime
16. Asymptotic sieve and the parity principle
17. Combinatorial identities
18. Asymptotic sieve for primes
19. Equidistribution of quadratic roots
20. Marching over Gaussian primes
21. Primes represented by polynomials
22. Level of distribution of arithmetic sequences
23. Primes in short intervals
24. The least prime in an arithmetic progression
25. Almost-prime sieve

This is a comprehensive and up-to-date treatment of sieve methods. The theory of the sieve is developed thoroughly with complete and accessible proofs of the basic theorems. Included is a wide range of applications, both to traditional questions such as those concerning primes, and to areas previously unexplored by sieve methods, such as elliptic curves, points on cubic surfaces and quantum ergodicity. New proofs are given also of some of the central theorems of analytic number theory; these proofs emphasize and take advantage of the applicability of sieve ideas.

The book contains numerous comments which provide the reader with insight into the workings of the subject, both as to what the sieve can do and what it cannot do. The authors reveal recent developments by which the parity barrier can be breached, exposing golden nuggets of the subject, previously inaccessible. The variety in the topics covered and in the levels of difficulty encountered makes this a work of value to novices and experts alike, both as an educational tool and a basic reference.