| Preface to the Dover Edition | p. iii |
| Preface | p. v |
| Notation | p. xi |
| Errata | p. xv |
| Introduction | p. 1 |
| Hypotheses H and HN | p. 1 |
| Sieve methods | p. 5 |
| Scope and presentation | p. 8 |
| The Sieve of Eratosthenes: Formulation of the General Sieve | p. 12 |
| Introductory remarks | p. 12 |
| The sequences A | p. 14 |
| Basic examples | p. 16 |
| The sifting set B and the sifting function S | p. 24 |
| The sieve of Eratosthenes-Legendre | p. 30 |
| The Combinatorial Sieve | p. 37 |
| The general method | p. 37 |
| Brun's pure sieve | p. 46 |
| Technical preparation | p. 52 |
| Brun's sieve | p. 56 |
| A general upper bound O-result | p. 68 |
| Sifting by a thin set of primes | p. 70 |
| Further applications | p. 75 |
| Fundamental Lemma | p. 82 |
| Rosser's sieve | p. 89 |
| The Simplest Selberg Upper Bound Method | p. 97 |
| The method | p. 97 |
| The case (d) = 1, Rd ≤ 1 | p. 101 |
| Application to 1 | p. 101 |
| The Brun-Titchmarsh inequality | p. 105 |
| The Titchmarsh divisor problem | p. 110 |
| The case (p) = p/p-1 | p. 113 |
| The prime twins and Goldbach problems: explicit upper bounds | p. 116 |
| The problem ap + b = p': an explicit upper bound | p. 119 |
| The Selberg Upper Bound Method (continued): O-results | p. 130 |
| A lower bound for G(x, z) | p. 130 |
| Applications | p. 133 |
| The Selberg Upper Bound Method: Explicit Estimates | p. 142 |
| A two-sided 2-condition | p. 142 |
| Technical preparation | p. 143 |
| Asymptotic formula for G(z) | p. 147 |
| The main theorems | p. 152 |
| Two ways of dealing with polynomial sequences {F(p)}: discussion | p. 153 |
| Primes representable by polynomials | p. 157 |
| Primes representable by polynomials F(p): the non-linearized approach | p. 167 |
| Prime k-tuplets | p. 172 |
| Primes representable by polynomials F(p): the linearized approach | p. 180 |
| An Extension of Selberg's Upper Bound Method | p. 187 |
| The method | p. 187 |
| An upper estimate | p. 191 |
| The function | p. 193 |
| Asymptotic formula for G(, z) | p. 197 |
| The main result | p. 202 |
| Selberg's Sieve Method (continued): A First Lower Bound | p. 204 |
| Combinatorial identities | p. 204 |
| Ah asymptotic formula for S | p. 206 |
| Fundamental Lemma | p. 208 |
| The function | p. 211 |
| A lower bound | p. 213 |
| The main result | p. 218 |
| TheLinear Sieve | p. 223 |
| The method | p. 223 |
| The functions F, f | p. 225 |
| An approximate identity for the leading terms | p. 228 |
| Upper and lower bounds for S | p. 231 |
| The main result | p. 236 |
| A Weighted Sieve: The Linear Case | p. 241 |
| The method | p. 241 |
| Application to the prime twins and Goldbach problems | p. 247 |
| The weighted sieve in applicable form | p. 252 |
| Almost-primes in intervals and arithmetic progressions | p. 256 |
| Almost-primes representable by irreducible polynomials F(n) | p. 259 |
| Almost-primes representable by irreducible polynomials F(p) | p. 261 |
| Weighted Sieves: The General Case | p. 269 |
| The first method | p. 269 |
| The first method in applicable form | p. 277 |
| Almost-primes representable by polynomials | p. 282 |
| The second method | p. 291 |
| Almost-primes representable by polynomials | p. 310 |
| Another method | p. 315 |
| Chen's Theorem | p. 320 |
| Introduction | p. 320 |
| The weighted sieve | p. 321 |
| Application of Selberg's upper sieve | p. 327 |
| Transition to primitive characters | p. 330 |
| Application of contour integration | p. 334 |
| Application of the large sieve | p. 336 |
| Bibliography | p. 339 |
| References | p. 342 |
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