Sieves in number theory

Includes index

Includes bibliography (p. 297-302) and references

1. The Structure of Sifting Arguments
1.1. The Sieves of Eratosthenes and Legendre
1.2. Examples of Sifting Situations
1.3. A General Formulation of a Sifting Situation
2. Selberg's Upper Bound Method
2.1. The Sifting Apparatus
2.2. General Estimates of G(x) and E(D, P)
2.3. Applications
3. Combinatorial Methods
3.1. The Construction of Combinatorial Sieves
3.2. Brun's Pure Sieve
3.3. A Modern Edition of Brun's Sieve
3.4. Brun's Version of his Method
4. Rosser's Sieve
4.1. Approximations by Continuous Functions
4.2. The Functions F and f
4.3. The Convergence Problem
4.4. A Sieve Theorem Following Rosser
4.5. Extremal Examples
5. The Sieve with Weights
5.1. Simpler Weighting Devices
5.2. More Elaborate Weighted Sieves
5.3. A Weighted Sieve Following Rosser
6. The Remainder Term in the Linear Sieve
6.1. The Bilinear Nature of Rosser's Construction
6.2. Sifting Short Intervals
7. Lower Bound Sieves when [kappa]> 1
7.1. An Extension of Selberg's Upper Bound
7.2. A Lower Bound Sieve via Buchstab's Identity
7.3. Selberg's [Lambda][Superscript 2] [Lambda] Method

This book surveys the current state of the "small" sieve methods developed by Brun, Selberg and later workers.. A self-contained treatment is given to topics that are of central importance in the subject. These include the upper bound method of Selberg, Brun's method, Rosser's sieve as developed by Iwaniec, with a bilinear form of the remainder term, the sieve with weights, and the use of Selberg's ideas in deriving lower-bound sieves. Further developments are introduced with the support of references t. The book is suitable for university graduates making their first acquaintance with the subject, leading them towards the frontiers of modern research and unsolved problems in the subject area.