| Introduction | p. 1 |
| The Structure of Sifting Arguments | p. 7 |
| The Sieves of Eratosthenes and Legendre | p. 7 |
| The Contribution of Eratosthenes | p. 7 |
| Legendre's Sieve | p. 8 |
| An Estimate for n(X) | p. 10 |
| The Distribution of Primes | p. 12 |
| Examples of Sifting Situations | p. 13 |
| Notations | p. 13 |
| The Integers in an Interval (Y - X, Y ) | p. 14 |
| Numbers Given by Polynomial Expressions | p. 15 |
| Arithmetic Progressions | p. 16 |
| Sums of Two Squares | p. 17 |
| Polynomials with Prime Arguments | p. 17 |
| A General Formulation of a Sifting Situation | p. 19 |
| The Basic Formulation | p. 19 |
| Legendre's Sieve in a General Setting | p. 23 |
| A Generalised Formulation | p. 25 |
| A Further Generalisation | p. 26 |
| Sifting Density | p. 27 |
| The Sifting Limit (k) | p. 37 |
| Composition of Sieves | p. 37 |
| Notes on Chapter 1 | p. 39 |
| Selberg's Upper Bound Method | p. 41 |
| The Sifting Apparatus | p. 42 |
| Selberg's Theorem | p. 42 |
| The Numbers (lambda)(d) | p. 46 |
| A Simple Application | p. 49 |
| General Estimates of G(x) and E(D, P) | p. 50 |
| An Estimate by Rankin's Device | p. 51 |
| Asymptotic Formulas | p. 54 |
| The Error Term | p. 60 |
| Applications | p. 63 |
| Arithmetic Progressions | p. 63 |
| Prime Twins and Goldbach's Problem | p. 64 |
| Polynomial Sequences | p. 66 |
| Notes on Chapter 2 | p. 68 |
| Combinatorial Methods | p. 71 |
| The Construction of Combinatorial Sieves | p. 72 |
| Preliminary Discussion of Brun's Ideas | p. 72 |
| Fundamental Inequalities and Identities | p. 73 |
| Buchstab's Identity | p. 77 |
| The Combinatorial Sieve Lemma | p. 78 |
| Brun's Pure Sieve | p. 79 |
| Inequalities and Identities | p. 80 |
| The "Pure Sieve" Theorem | p. 81 |
| A Corollary | p. 83 |
| Prime Twins | p. 84 |
| A Modern Edition of Brun's Sieve | p. 85 |
| Rosser's Choice of X | p. 86 |
| A Technical Estimate | p. 87 |
| A Simplifying Approximation | p. 88 |
| A Combinatorial Sieve Theorem | p. 89 |
| Applications | p. 94 |
| Brun's Version of his Method | p. 96 |
| Brun's Choice of x | p. 96 |
| The Estimations | p. 97 |
| The Result | p. 98 |
| Notes on Chapter 3 | p. 100 |
| Rosser's Sieve | p. 103 |
| Approximations by Continuous Functions | p. 105 |
| The Recurrence Relations | p. 106 |
| Partial Summation | p. 108 |
| The Leading Terms | p. 110 |
| The Functions F and f | p. 113 |
| The Difference-Differential Equations | p. 113 |
| The Adjoint Equation and the Inner Product | p. 115 |
| Solutions of the Adjoint Equation | p. 117 |
| Particular Values of F(s) and f(s) | p. 124 |
| Asymptotic Analysis as k -> $(infinity$) | p. 129 |
| The Convergence Problem | p. 134 |
| The Auxiliary Functions | p. 135 |
| Adjoints and Inner Products | p. 137 |
| The Case k </= 1/2 | p. 141 |
| A Sieve Theorem Following Rosser | p. 144 |
| The Case k >/= 1/2: a First Result | p. 145 |
| Theorem 1 when k</= 1/2 | p. 148 |
| An Improved Version of Proposition 1 | p. 150 |
| A Two-Sided Estimate | p. 156 |
| Extremal Examples | p. 158 |
| The Linear Case | p. 158 |
| The Case k=1/2 | p. 164 |
| Notes on Chapter 4 | p. 167 |
| The Sieve with Weights | p. 173 |
| Simpler Weighting Devices | p. 176 |
| Logarithmic Weights | p. 176 |
| Modified Logarithmic Weights | p. 179 |
| Some Applications | p. 184 |
| More Elaborate Weighted Sieves | p. 185 |
| An Improved Weighting Device | p. 186 |
| Buchstab's Weights | p. 188 |
| A Weighted Sieve Following Rosser | p. 193 |
| Combining Sieving and Weighting | p. 194 |
| The Reduction Identities | p. 202 |
| An Identity for the Main Term | p. 205 |
| The Estimate for the Main Term | p. 210 |
| Notes on Chapter 5 | p. 216 |
| The Remainder Term in the Linear Sieve | p. 223 |
| The Bilinear Nature of Rosser's Construction | p. 226 |
| The Factorisation of x.d | p. 227 |
| Discretisations of Rosser's Sieve | p. 229 |
| Specification of Details | p. 234 |
| The Leading Contributions to the Main Term | p. 238 |
| Composition of Sieves | p. 241 |
| The Remainder Term | p. 243 |
| Sifting Short Intervals | p. 245 |
| The Smoothed Formulation | p. 246 |
| The Remainder Sums | p. 250 |
| Trigonometrical Sums | p. 252 |
| Notes on Chapter 6 | p. 256 |
| Lower Bound Sieves when k > 1 | p. 259 |
| An Extension of Selberg's Upper Bound | p. 260 |
| The Integral Equation and the Function $(sigma$) (s) | p. 261 |
| The Estimation of G(s) | p. 265 |
| A Lower Bound Sieve via Buchstab's Identity | p. 270 |
| Buchstab's Iterations | p. 271 |
| The Buchstab Transform of the $(lambda$)2 Method | p. 273 |
| The Sifting Limit as k -> $(infinity$) | p. 277 |
| Selberg's a2 a" Method | p. 285 |
| An Identity for the Main Term | p. 286 |
| The Improved Sifting Limit for Large k | p. 288 |
| Notes on Chapter 7 | p. 291 |
| References | p. 297 |
| Index | p. 303 |
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