Sieve Methods Book

Preface to the Dover Edition iii

Preface v

Notation xi

Errata xv

Introduction 1

1 Hypotheses H and H_N 1

2 Sieve methods 5

3 Scope and presentation 8

1 The Sieve of Eratosthenes: Formulation of the General Sieve 12

1 Introductory remarks 12

2 The sequences A 14

3 Basic examples 16

4 The sifting set B and the sifting function S 24

5 The sieve of Eratosthenes-Legendre 30

2 The Combinatorial Sieve 37

1 The general method 37

2 Brun's pure sieve 46

3 Technical preparation 52

4 Brun's sieve 56

5 A general upper bound O-result 68

6 Sifting by a thin set of primes 70

7 Further applications 75

8 Fundamental Lemma 82

9 Rosser's sieve 89

3 The Simplest Selberg Upper Bound Method 97

1 The method 97

2 The case ω(d) = 1, |R_d| ≤ 1 101

3 Application to 1 101

4 The Brun-Titchmarsh inequality 105

5 The Titchmarsh divisor problem 110

6 The case ω(p) = p/p-1 113

7 The prime twins and Goldbach problems: explicit upper bounds 116

8 The problem ap + b = p': an explicit upper bound 119

4 The Selberg Upper Bound Method (continued): O-results 130

1 A lower bound for G(x, z) 130

2 Applications 133

5 The Selberg Upper Bound Method: Explicit Estimates 142

1 A two-sided Ω₂-condition 142

2 Technical preparation 143

3 Asymptotic formula for G(z) 147

4 The main theorems 152

5 Two ways of dealing with polynomial sequences {F(p)}: discussion 153

6 Primes representable by polynomials 157

7 Primes representable by polynomials F(p): the non-linearized approach 167

8 Prime k-tuplets 172

9 Primes representable by polynomials F(p): the linearized approach 180

6 An Extension of Selberg's Upper Bound Method 187

1 The method 187

2 An upper estimate 191

3 The function σ_κ 193

4 Asymptotic formula for G(ξ, z) 197

5 The main result 202

7 Selberg's Sieve Method (continued): A First Lower Bound 204

1 Combinatorial identities 204

2 Ah asymptotic formula for S 206

3 Fundamental Lemma 208

4 The function η_κ 211

5 A lower bound 213

6 The main result 218

8 TheLinear Sieve 223

1 The method 223

2 The functions F, f 225

3 An approximate identity for the leading terms 228

4 Upper and lower bounds for S 231

5 The main result 236

9 A Weighted Sieve: The Linear Case 241

1 The method 241

2 Application to the prime twins and Goldbach problems 247

3 The weighted sieve in applicable form 252

4 Almost-primes in intervals and arithmetic progressions 256

5 Almost-primes representable by irreducible polynomials F(n) 259

6 Almost-primes representable by irreducible polynomials F(p) 261

10 Weighted Sieves: The General Case 269

1 The first method 269

2 The first method in applicable form 277

3 Almost-primes representable by polynomials 282

4 The second method 291

5 Almost-primes representable by polynomials 310

6 Another method 315

11 Chen's Theorem 320

1 Introduction 320

2 The weighted sieve 321

3 Application of Selberg's upper sieve 327

4 Transition to primitive characters 330

5 Application of contour integration 334

6 Application of the large sieve 336

Bibliography 339

References 342