Prime-detecting sieves / Glyn Harman.

Format:

Book

Author/Creator:

Harman, G. (Glyn), 1956-

Series:

London Mathematical Society monographs ; new ser., no. 33.

London Mathematical Society monographs

Language:

English

Subjects (All):

Sieves (Mathematics).

Numbers, Prime.

Number theory.

Physical Description:

xiv, 362 pages : illustrations ; 24 cm.

Place of Publication:

Princeton : Princeton University Press, [2007]

Summary:

This book seeks to describe the rapid development in recent decades of sieve methods able to detect prime numbers. The subject began with Eratosthenes in antiquity, took on new shape with Legendre's form of the sieve, was substantially reworked by Ivan M. Vinogradov and Yuri V. Linnik, but came into its own with Robert C. Vaughan and important contributions from others, notably Roger Heath-Brown and Henryk Iwaniec. Prime-Detecting Sieves breaks new ground by bringing together several different types of problems that have been tackled with modern sieve methods and by discussing the ideas common to each, in particular the use of Type I and Type II information.

No other book has undertaken such a systematic treatment of prime-detecting sieves. Among the many topics Glyn Harman covers are primes in short intervals, the greatest prime factor of the sequence of shifted primes, Goldbach numbers in short intervals, the distribution of Gaussian primes, and the recent work of John Friedlander and Iwaniec on primes that are a sum of a square and a fourth power, and Heath-Brown's work on primes represented as a cube plus twice a cube. This book contains much that is accessible to beginning graduate students, yet also provides insights that will benefit established researchers.

Contents:

1.1 The Beginning 1

1.2 The Sieve of Eratosthenes 4

1.3 The Sieve of Eratosthenes-Legendre 6

1.4 The Prime Number Theorem and Its Consequences 8

1.5 Brun, Selberg, and Rosser-Iwaniec 18

1.6 Eratosthenes-Legendre-Vinogradov 20

Chapter 2 The Vaughan Identity 25

2.2 An Exponential Sum over Primes 28

2.3 The Distribution of [alpha]p Modulo 1 29

2.4 The Bombieri-Vinogradov Theorem 33

2.5 Linnik's and Heath-Brown's Identities 38

2.6 Further Thoughts on Vaughan's Identity 42

Chapter 3 The Alternative Sieve 47

3.2 Cosmetic Surgery 49

3.3 The Fundamental Theorem 50

3.4 Application to the Distribution of {[alpha]p} 54

3.5 A Lower-Bound Sieve 56

3.6 A Change of Notation 60

3.7 The Piatetski-Shapiro PNT 62

3.8 Historical Note 63

Chapter 4 The Rosser-Iwaniec Sieve 65

4.2 A Fundamental Lemma 67

4.3 A Heuristic Argument 72

4.4 Proof of the Lower-Bound Sieve 73

4.5 Developments of the Rosser-Iwaniec Sieve 79

Chapter 5 Developing the Alternative Sieve 83

5.2 New Forms of the Fundamental Theorem 83

5.3 Reversing Roles 86

5.4 A New Idea 91

5.5 Higher-Dimensional Versions 92

5.6 Greatest Prime Factors 93

Chapter 6 An Upper-Bound Sieve 103

6.1 The Method Described 103

6.2 A Device by Chebychev 105

6.3 The Arithmetical Information 107

6.4 Applying the Rosser-Iwaniec Sieve 110

6.5 An Asymptotic Formula 112

6.6 The Alternative Sieve Applied 112

6.7 Upper-Bounds: Region by Region 115

6.8 Why a Previous Idea Fails 118

Chapter 7 Primes in Short Intervals 119

7.1 The Zero-Density Approach 119

7.2 Preliminary Results 121

7.3 The 7/12 Result 128

7.4 Shorter Intervals 133

7.5 Application of Watt's Theorem 135

7.6 Sieve Asymptotic Formulae 139

7.7 The Two-Dimensional Sieve Revisited 143

7.8 Further Asymptotic Formulae 147

7.9 The Final Decomposition 150

7.10 Where to Now? 155

Chapter 8 The Brun-Titchmarsh Theorem on Average 157

8.2 The Arithmetical Information 159

8.3 The Alternative Sieve Applied 165

8.4 The Alternative Sieve for [tau less than or equal alpha]1 [less than or equal] 3/7, [theta] [less than or equal] 11/21 172

8.5 The Alternative Sieve in Two Dimensions 174

8.6 The Alternative Sieve in Three Dimensions 178

8.7 An Upper Bound for Large [theta] 182

8.8 Completion of Proof 183

Chapter 9 Primes in Almost All Intervals 189

9.2 The Arithmetical Information 191

9.3 The Alternative Sieve Applied 195

9.4 The Final Decomposition 198

9.5 An Upper-Bound Result 199

9.6 Other Measures of Gaps Between Primes 200

Chapter 10 Combination with the Vector Sieve 201

10.2 Goldbach Numbers in Short Intervals 202

10.3 Proof of Theorem 205

10.4 Dirichlet Polynomials 211

10.5 Sieving the Interval B[subscript 1] 218

10.6 Sieving the Interval B[subscript 2] 227

10.7 Further Applications 229

Chapter 11 Generalizing to Algebraic Number Fields 231

11.2 Gaussian Primes in Sectors 232

11.3 Notation and Outline of the Method 233

11.4 The Arithmetical Information 237

11.5 Asymptotic Formulae for Problem 1 240

11.6 The Final Decomposition for Problem 1 244

11.7 Prime Ideals in Small Regions 247

11.9 Estimates for Dirichlet Polynomials 255

11.10 Asymptotic Formulae for Problem 2 258

11.11 The Final Decomposition for Problem 2 260

Chapter 12 Variations on Gaussian Primes 265

12.2 Outline of the Fouvry-Iwaniec Method 266

12.3 Some Preliminary Results 268

12.4 Fouvry-Iwaniec Type I Information 273

12.5 Reducing the Bilinear Form Problem 276

12.6 Catching the Cancellation Introduced by [Mu] 279

12.7 The Main Term for Theorem 12.1 284

12.8 The Friedlander-Iwaniec Outline for a[superscript a] + b[superscript 4] 285

12.9 The Friedlander-Iwaniec Asymptotic Sieve 287

12.10 Sketch of the Crucial Result 296

12.11 And Now? 301

Chapter 13 Primes of the Form x[superscript 3] + 2y[superscript 3] 303

13.2 Outline of the Proof 304

13.3 Preliminary Results 312

13.4 The Type I Estimates 313

13.5 The Fundamental Lemma Result 316

13.6 Proof of Lemma 13.6 317

13.7 Proof of Lemma 13.7 325

13.8 The Type II Information Established 326

14.2 A Challenge with Which to Close 336

A.1 Perron's formula 337

A.2 Buchstab's Function [omega](u) 339

A.3 Large-Sieve Inequalities 343

A.4 The Mean Value Theorem for Dirichlet Polynomials 346

A.5 Smooth Functions 347.

Notes:

Includes bibliographical references (pages [349]-359) and index.

ISBN:

069112437X

9780691124377

OCLC:

132585714

Online:

Publisher description

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