Bounded gaps between primes: the epic breakthroughs of the early twenty-first century
"Searching for small gaps between consecutive primes is one way to approach the twin primes conjecture, one of the most celebrated unsolved problems in number theory. This book documents the remarkable developments of recent decades, whereby an upper bound on the known gap length between infini...
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge, United Kingdom
Cambridge University Press
2021
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "Searching for small gaps between consecutive primes is one way to approach the twin primes conjecture, one of the most celebrated unsolved problems in number theory. This book documents the remarkable developments of recent decades, whereby an upper bound on the known gap length between infinite numbers of consecutive primes has been reduced to a tractable finite size. The text is both introductory and complete: the detailed way in which results are proved is fully set out and plenty of background material is included. The reader journeys from selected historical theorems to the latest best result, exploring the contributions of a vast array of mathematicians, including Bombieri, Goldston, Motohashi, Pintz, Yildirim, Zhang, Maynard, Tao and Polymath8. The book is supported by a linked and freely-available package of computer programs. The material is suitable for graduate students and of interest to any mathematician curious about recent breakthroughs in the field"-- |
| Beschreibung: | xiv, 576 Seiten Illustration, Diagramme |
| ISBN: | 9781108836746 9781108799201 |
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Datensatz im Suchindex
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| adam_text | Contents Preface Acknowledgements 2 page xi xiv Introduction Why This Study? 1.1 Summary of This Chapter 1.2 History and Overview of These Developments 1.3 1.4 Polymath Projects and Members of Polymath8 Timeline of Developments 1.5 Prime Patterns and the Hardy-Littlewood Conjecture 1.6 Jumping Champions 1.7 The von Mangoldt Function 1.8 The В ombieri-Vinogradov Theorem 1.9 1.10 Admissible Tuples 1.10.1 Introduction 1.10.2 Bounds for H{k) 1.10.3 The Second Hardy-Littlewood Conjecture 1.11 A Brief Guide to the Literature 1.12 End Notes 1 1 2 2 5 8 9 13 15 18 21 21 24 26 31 33 The Sieves of Brun and Selberg Introduction 2.1 Summary of This Chapter 2.2 Bran’s Pure Sieve 2.3 Bran’s Pure Sieve Addendum 2.4 The Selberg Sieve 2.5 Making the Constant Explicit 2.6 An Application to a Bran-Titchmarsh Inequality 2.7 Bran’s, Selberg’s and Other Sieves 2.8 35 35 36 38 46 47 60 70 72 vii
viii Contents 2.9 2.10 A Brief Reader’s Guide to Sieve Theory End Note: Twin Almost Primes and the Sieve Parity Problem 82 83 3 Early Work 3.1 Introduction 3.2 Chapter Summary 3.3 Erdős and the First Unconditional Step 3.4 The Beautiful Method of Bombieri and Davenport 3.5 Maier’s Matrix Method 3.6 End Notes 89 89 90 91 93 112 113 4 The Breakthrough of Goldston, Motohashi, Pintz and Yildirim 4.1 Introduction 4.2 Outline of the GPY Method 4.3 Definitions and Summary 4.4 General Preliminary Results 4.5 Special Preliminary Results 4.6 The Essential Theorem of Gallagher 4.7 The Main GPY Theorem 4.8 The Simplified Proof 4.9 GPY’s Conditional Bounded Gaps Theorem 4.10 End Notes 114 114 116 120 127 135 158 167 168 180 183 5 The Astounding Result of Yitang Zhang 5.1 Introduction 5.2 Summary of Zhang’s Method 5.3 Notation 5.4 Chapter Summary 5.5 Variations on the Bombieri-Vinogradov Estimates 5.6 Preliminary Lemmas 5.7 Upper Bound for the Sum Si 5.8 Lower Bound for the Sum Տշ 5.9 Zhang’s Prime Gap Result 5.10 End Notes 184 184 187 189 189 190 192 200 210 216 217 6 Maynard’s Radical Simplification 6.1 Introduction 6.2 Definitions 6.3 Chapter Summary 6.4 Selberg’s Sieve Lemmas 6.5 Other Preliminary Lemmas 6.6 Fundamental Lemmas 219 219 221 223 226 238 251
Contents 6.7 6.8 6.9 6.10 Integration Formulas Maynard’s Algorithm Main Theorems End Notes ix 261 265 266 271 7 Polymath’s Refinements of Maynard’s Results 7.1 Introduction 7.2 Definitions 7.3 Chapter Summary 7.4 Preliminary Results 7.5 Polymath’s Algorithm for M* 7.6 Limits to These Techniques: Upper Bound for M* 7.7 Bogáért’s Krylov Basis Method 7.8 Bogáért’s Algorithm 7.9 How the Gap Bound pn+ — pn 246 Is Derived 7.10 Limits to This Approach for M*,e 7.11 End Notes 272 272 274 275 279 300 302 305 312 314 320 323 8 Variations on Bombieri-Vinogradov 8.1 Introduction 8.2 Special Notations and Definitions 8.3 Chapter Summary 8.4 Preliminary Results 8.5 Multiple Dense Divisibility 8.6 Improving Zhang 8.7 A Fundamental Technical Result 8.8 Using Heath-Brown’s Identity 8.9 One-Dimensional Exponential Sums 8.10 Polymath’s Type I and II Estimates 8.11 Application to Prime Gaps 8.12 End Notes 327 327 328 332 337 343 347 377 385 399 433 444 446 9 Further Work and the Epilogue 9.1 Introduction 9.2 Assuming Elliott-Halberstam’s Conjecture 9.3 Assuming the Generalized Elliott-Halberstam Conjecture 9.4 Gaps between Almost Primes 9.5 Affine Forms and Clusters of Primes inIntervals 9.6 Limit Points of Normalized ConsecutivePrime Differences 9.7 Artin’s Primitive Root Conjecture 9.8 Consecutive Primes in AP with a FixedCommon Difference 451 451 451 452 452 454 455 456 457
x Contents 9.9 9.10 9.11 9.12 Prime Ideals and IrreduciblePolynomials Coefficients of Modular Forms Elliptic Curves Epilogue 457 458 459 460 Appendix A Bessel Functions of the First Kind 465 Appendix В A Type of Compact Symmetric Operator 473 Appendix C Solving an Optimization Problem 480 Appendix D A Brun-Titchmarsh Inequality 492 Appendix E The Weil Exponential Sum Bound 502 Appendix F Complex Function Theory 516 Appendix G The Dispersion Method of Linnik 522 Appendix H One Thousand Admissible Tuples 528 Appendix I PGpack Minimanual 531 References Index 555 567
|
| adam_txt |
Contents Preface Acknowledgements 2 page xi xiv Introduction Why This Study? 1.1 Summary of This Chapter 1.2 History and Overview of These Developments 1.3 1.4 Polymath Projects and Members of Polymath8 Timeline of Developments 1.5 Prime Patterns and the Hardy-Littlewood Conjecture 1.6 Jumping Champions 1.7 The von Mangoldt Function 1.8 The В ombieri-Vinogradov Theorem 1.9 1.10 Admissible Tuples 1.10.1 Introduction 1.10.2 Bounds for H{k) 1.10.3 The Second Hardy-Littlewood Conjecture 1.11 A Brief Guide to the Literature 1.12 End Notes 1 1 2 2 5 8 9 13 15 18 21 21 24 26 31 33 The Sieves of Brun and Selberg Introduction 2.1 Summary of This Chapter 2.2 Bran’s Pure Sieve 2.3 Bran’s Pure Sieve Addendum 2.4 The Selberg Sieve 2.5 Making the Constant Explicit 2.6 An Application to a Bran-Titchmarsh Inequality 2.7 Bran’s, Selberg’s and Other Sieves 2.8 35 35 36 38 46 47 60 70 72 vii
viii Contents 2.9 2.10 A Brief Reader’s Guide to Sieve Theory End Note: Twin Almost Primes and the Sieve Parity Problem 82 83 3 Early Work 3.1 Introduction 3.2 Chapter Summary 3.3 Erdős and the First Unconditional Step 3.4 The Beautiful Method of Bombieri and Davenport 3.5 Maier’s Matrix Method 3.6 End Notes 89 89 90 91 93 112 113 4 The Breakthrough of Goldston, Motohashi, Pintz and Yildirim 4.1 Introduction 4.2 Outline of the GPY Method 4.3 Definitions and Summary 4.4 General Preliminary Results 4.5 Special Preliminary Results 4.6 The Essential Theorem of Gallagher 4.7 The Main GPY Theorem 4.8 The Simplified Proof 4.9 GPY’s Conditional Bounded Gaps Theorem 4.10 End Notes 114 114 116 120 127 135 158 167 168 180 183 5 The Astounding Result of Yitang Zhang 5.1 Introduction 5.2 Summary of Zhang’s Method 5.3 Notation 5.4 Chapter Summary 5.5 Variations on the Bombieri-Vinogradov Estimates 5.6 Preliminary Lemmas 5.7 Upper Bound for the Sum Si 5.8 Lower Bound for the Sum Տշ 5.9 Zhang’s Prime Gap Result 5.10 End Notes 184 184 187 189 189 190 192 200 210 216 217 6 Maynard’s Radical Simplification 6.1 Introduction 6.2 Definitions 6.3 Chapter Summary 6.4 Selberg’s Sieve Lemmas 6.5 Other Preliminary Lemmas 6.6 Fundamental Lemmas 219 219 221 223 226 238 251
Contents 6.7 6.8 6.9 6.10 Integration Formulas Maynard’s Algorithm Main Theorems End Notes ix 261 265 266 271 7 Polymath’s Refinements of Maynard’s Results 7.1 Introduction 7.2 Definitions 7.3 Chapter Summary 7.4 Preliminary Results 7.5 Polymath’s Algorithm for M* 7.6 Limits to These Techniques: Upper Bound for M* 7.7 Bogáért’s Krylov Basis Method 7.8 Bogáért’s Algorithm 7.9 How the Gap Bound pn+\ — pn 246 Is Derived 7.10 Limits to This Approach for M*,e 7.11 End Notes 272 272 274 275 279 300 302 305 312 314 320 323 8 Variations on Bombieri-Vinogradov 8.1 Introduction 8.2 Special Notations and Definitions 8.3 Chapter Summary 8.4 Preliminary Results 8.5 Multiple Dense Divisibility 8.6 Improving Zhang 8.7 A Fundamental Technical Result 8.8 Using Heath-Brown’s Identity 8.9 One-Dimensional Exponential Sums 8.10 Polymath’s Type I and II Estimates 8.11 Application to Prime Gaps 8.12 End Notes 327 327 328 332 337 343 347 377 385 399 433 444 446 9 Further Work and the Epilogue 9.1 Introduction 9.2 Assuming Elliott-Halberstam’s Conjecture 9.3 Assuming the Generalized Elliott-Halberstam Conjecture 9.4 Gaps between Almost Primes 9.5 Affine Forms and Clusters of Primes inIntervals 9.6 Limit Points of Normalized ConsecutivePrime Differences 9.7 Artin’s Primitive Root Conjecture 9.8 Consecutive Primes in AP with a FixedCommon Difference 451 451 451 452 452 454 455 456 457
x Contents 9.9 9.10 9.11 9.12 Prime Ideals and IrreduciblePolynomials Coefficients of Modular Forms Elliptic Curves Epilogue 457 458 459 460 Appendix A Bessel Functions of the First Kind 465 Appendix В A Type of Compact Symmetric Operator 473 Appendix C Solving an Optimization Problem 480 Appendix D A Brun-Titchmarsh Inequality 492 Appendix E The Weil Exponential Sum Bound 502 Appendix F Complex Function Theory 516 Appendix G The Dispersion Method of Linnik 522 Appendix H One Thousand Admissible Tuples 528 Appendix I PGpack Minimanual 531 References Index 555 567 |
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| spelling | Broughan, Kevin A. 1943- Verfasser (DE-588)1137861630 aut Bounded gaps between primes the epic breakthroughs of the early twenty-first century Kevin Broughan (University of Waikoato) Cambridge, United Kingdom Cambridge University Press 2021 © 2021 xiv, 576 Seiten Illustration, Diagramme txt rdacontent n rdamedia nc rdacarrier "Searching for small gaps between consecutive primes is one way to approach the twin primes conjecture, one of the most celebrated unsolved problems in number theory. This book documents the remarkable developments of recent decades, whereby an upper bound on the known gap length between infinite numbers of consecutive primes has been reduced to a tractable finite size. The text is both introductory and complete: the detailed way in which results are proved is fully set out and plenty of background material is included. The reader journeys from selected historical theorems to the latest best result, exploring the contributions of a vast array of mathematicians, including Bombieri, Goldston, Motohashi, Pintz, Yildirim, Zhang, Maynard, Tao and Polymath8. The book is supported by a linked and freely-available package of computer programs. The material is suitable for graduate students and of interest to any mathematician curious about recent breakthroughs in the field"-- Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Primzahltheorie (DE-588)4175715-4 gnd rswk-swf Numbers, Prime Number theory Zahlentheorie (DE-588)4067277-3 s Primzahltheorie (DE-588)4175715-4 s DE-604 Erscheint auch als Online-Ausgabe, EPUB 10.1017/9781108872201 978-1-108-87220-1 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032738786&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Broughan, Kevin A. 1943- Bounded gaps between primes the epic breakthroughs of the early twenty-first century Zahlentheorie (DE-588)4067277-3 gnd Primzahltheorie (DE-588)4175715-4 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4175715-4 |
| title | Bounded gaps between primes the epic breakthroughs of the early twenty-first century |
| title_auth | Bounded gaps between primes the epic breakthroughs of the early twenty-first century |
| title_exact_search | Bounded gaps between primes the epic breakthroughs of the early twenty-first century |
| title_exact_search_txtP | Bounded gaps between primes the epic breakthroughs of the early twenty-first century |
| title_full | Bounded gaps between primes the epic breakthroughs of the early twenty-first century Kevin Broughan (University of Waikoato) |
| title_fullStr | Bounded gaps between primes the epic breakthroughs of the early twenty-first century Kevin Broughan (University of Waikoato) |
| title_full_unstemmed | Bounded gaps between primes the epic breakthroughs of the early twenty-first century Kevin Broughan (University of Waikoato) |
| title_short | Bounded gaps between primes |
| title_sort | bounded gaps between primes the epic breakthroughs of the early twenty first century |
| title_sub | the epic breakthroughs of the early twenty-first century |
| topic | Zahlentheorie (DE-588)4067277-3 gnd Primzahltheorie (DE-588)4175715-4 gnd |
| topic_facet | Zahlentheorie Primzahltheorie |
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