As probability and combinatorics have penetrated the fabric of mathematical activity, sieve methods have become more versatile and sophisticated and in recent years have played a part in some of the most spectacular mathematical discoveries. Nearly a hundred years have passed since Viggo Brun invented his famous sieve, and the use of sieve methods is constantly evolving. Many arithmetical investigations encounter a combinatorial problem that requires a sieving argument, and this tract offers a modern and reliable guide in such situations. The theory of higher dimensional sieves is thoroughly explored, and examples are provided throughout. A Mathematica® software package for sieve-theoretical calculations is provided on the authors' website. To further benefit readers, the Appendix describes methods for computing sieve functions.
Author(s): Harold G. Diamond, H. Halberstam, William F. Galway
Series: Cambridge Tracts in Mathematics
Publisher: Cambridge University Press
Year: 2008
Language: English
Pages: 290
Cover......Page 1
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
List of Illustrations......Page 13
List of Tables......Page 15
Preface......Page 17
Standard terminology......Page 19
Summatory functions......Page 20
Sifting functions......Page 21
Constants/parameters......Page 22
"Modifying" functions......Page 23
Part I Sieves......Page 25
1.1 The sieve problem......Page 27
1.2 Some basic hypotheses......Page 28
1.3 Prime g-tuples......Page 30
1.4 The Omega(Kappa) condition......Page 32
1.5 Notes on Chapter 1......Page 35
2.1 Improving the Eratosthenes—Legendre sieve......Page 37
2.2 A new parameter......Page 38
2.3 Notes on Chapter 2......Page 41
3.1 The fundamental sieve identity......Page 43
3.2 Efficacy of the Selberg sieve......Page 46
3.3 Multiplicative structure of modifying functions......Page 49
3.4 Notation: P, S(A,P,z), and V......Page 50
3.5 Notes on Chapter 3......Page 51
4.1 A start: an asymptotic formula for S(Aq, p, z)......Page 53
4.2 A lower bound for S(Aq, p, z)......Page 57
4.3 Notes on Chapter 4......Page 66
5.1 A lower bound for G(Xi, z)......Page 67
5.2 Asymptotics for G*(Xi, z)......Page 76
5.3 The j and Sigma functions......Page 80
5.4 Prime values of polynomials......Page 88
5.5 Notes on Chapter 5......Page 90
6.1 Statement of the main analytic theorem......Page 91
6.2 The S(x) functions......Page 94
6.3 The "linear" case Kappa = 1......Page 95
6.4 The cases Kappa > 1......Page 97
6.5 Notes on Chapter 6......Page 103
7.1 The theorem and first steps......Page 105
7.2 Bounds for V…......Page 109
7.3 Bounds for V…......Page 112
7.4 Completion of the proof of Theorem 7.1......Page 116
7.5 Notes on Chapter 7......Page 119
8.1 Toward the twin prime conjecture......Page 121
8.2 Notes on Chapter 8......Page 126
9.1 The main theorem and start of the proof......Page 127
9.2 The S21 and S22 sums......Page 131
9.3 Bounds on…......Page 134
9.4 Completion of the proof of Theorem 9.1......Page 144
9.5 Notes on Chapter 9......Page 146
10.1 A Mertens-type approximation......Page 149
10.2 The sieve setup and examples......Page 153
11.1 Introduction and additional conditions......Page 159
11.2 A set of weights......Page 161
11.3 Arithmetic interpretation......Page 164
11.4 A simple estimate......Page 168
11.5 Products of irreducible polynomials......Page 171
11.6 Polynomials at prime arguments......Page 173
11.7 Other weights......Page 174
11.8 Notes on Chapter 11......Page 175
Part II Proof of the Main Analytic Theorem......Page 177
12.1 P and Q and their adjoints......Page 179
12.2 Rapidly vanishing functions......Page 182
12.3 The Pi and Xi functions......Page 184
12.4 Notes on Chapter 12......Page 185
13.1 Two different sieve situations......Page 187
13.2 A necessary condition......Page 188
13.3 A program for determining F and f......Page 190
14.1 Lower bounds on Sigma......Page 193
14.2 Differential relations......Page 197
14.3 The adjoint function of j......Page 201
14.4 Inequalities for 1 — j......Page 202
14.5 Relations between Sigma' and 1 — Sigma......Page 207
14.6 The Xi function......Page 208
14.8 Notes on Chapter 14......Page 214
15.1 The p functions......Page 217
15.2 The q functions......Page 219
15.3 Zeros of the q functions......Page 220
15.4 Monotonicity and convexity relations......Page 221
15.5 Some lower bounds for pKappa......Page 223
15.6 An upper bound for pKappa......Page 225
15.7 The integrands of…......Page 226
16.1 Properties of the Pi and Xi functions......Page 231
16.2 Solution of some Pi and Xi equations......Page 233
16.3 Estimation of…......Page 238
17.1 The cases Kappa = 1, 1.5......Page 241
17.2 The cases Kappa = 2, 2.5, 3 ,…......Page 244
17.3 Proof of Proposition 17.3......Page 246
17.4 Notes on Chapter 17......Page 251
18.2 QKappa(u) > 0 for u > 0......Page 253
Appendix 1 Procedures for computing sieve functions......Page 257
A1.1 DDEs and the Iwaniec inner product......Page 258
A1.2 The upper and lower bound sieve functions......Page 259
A1.3 Using the Iwaniec inner product......Page 260
A1.4 Some features of Mathematica......Page 263
A1.5 Computing FKappa(u) and fKappa(u)......Page 264
A1.6 The function Ein(z)......Page 265
A1.7 Computing the adjoint functions......Page 266
A1.8 Computing jKappa(u)......Page 274
A1.9 Computing AlphaKappa and BetaKappa......Page 278
A1.10 Weighted-sieve computations......Page 279
Bibliography......Page 283
Index......Page 289
