An invitation to modern number theory:
In a manner accessible to beginning undergraduates, An Invitation to Modern Number Theory introduces many of the central problems, conjectures, results, and techniques of the field, such as the Riemann Hypothesis, Roth's Theorem, the Circle Method, and Random Matrix Theory. Showing how experime...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Princeton [u.a.]
Princeton Univ. Press
2006
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | In a manner accessible to beginning undergraduates, An Invitation to Modern Number Theory introduces many of the central problems, conjectures, results, and techniques of the field, such as the Riemann Hypothesis, Roth's Theorem, the Circle Method, and Random Matrix Theory. Showing how experiments are used to test conjectures and prove theorems, the book allows students to do original work on such problems, often using little more than calculus (though there are numerous remarks for those with deeper backgrounds). It shows students what number theory theorems are used for and what led to them and suggests problems for further research. Steven Miller and Ramin Takloo-Bighash introduce the problems and the computational skills required to numerically investigate them, providing background material (from probability to statistics to Fourier analysis) whenever necessary. They guide students through a variety of problems, ranging from basic number theory, cryptography, and Goldbach's Problem, to the algebraic structures of numbers and continued fractions, showing connections between these subjects and encouraging students to study them further. In addition, this is the first undergraduate book to explore Random Matrix Theory, which has recently become a powerful tool for predicting answers in number theory. Providing exercises, references to the background literature, and Web links to previous student research projects, An Invitation to Modern Number Theory can be used to teach a research seminar or a lecture class. |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XX, 503 S. |
| ISBN: | 0691120609 9780691120607 |
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| 520 | 3 | |a In a manner accessible to beginning undergraduates, An Invitation to Modern Number Theory introduces many of the central problems, conjectures, results, and techniques of the field, such as the Riemann Hypothesis, Roth's Theorem, the Circle Method, and Random Matrix Theory. Showing how experiments are used to test conjectures and prove theorems, the book allows students to do original work on such problems, often using little more than calculus (though there are numerous remarks for those with deeper backgrounds). It shows students what number theory theorems are used for and what led to them and suggests problems for further research. Steven Miller and Ramin Takloo-Bighash introduce the problems and the computational skills required to numerically investigate them, providing background material (from probability to statistics to Fourier analysis) whenever necessary. They guide students through a variety of problems, ranging from basic number theory, cryptography, and Goldbach's Problem, to the algebraic structures of numbers and continued fractions, showing connections between these subjects and encouraging students to study them further. In addition, this is the first undergraduate book to explore Random Matrix Theory, which has recently become a powerful tool for predicting answers in number theory. Providing exercises, references to the background literature, and Web links to previous student research projects, An Invitation to Modern Number Theory can be used to teach a research seminar or a lecture class. | |
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| adam_text | AN INVITATION TO MODERN NUMBER THEORY STEVEN J. MILLER AND RAMIN
TAKLOO-BIGHASH PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD CONTENTS
FOREWORD XI PREFACE XIII NOTATION XIX PART 1. BASIC NUMBER THEORY 1
CHAPTER 1. MOD P ARITHMETIC, GROUP THEORY AND CRYPTOGRAPHY 3 1.1
CRYPTOGRAPHY 3 1.2 EFFICIENT ALGORITHMS 5 1.3 CLOCK ARITHMETIC:
ARITHMETIC MODULO N 14 1.4 GROUP THEORY 15 1.5 RSA REVISITED 20 1.6
EISENSTEIN S PROOF OF QUADRATIC RECIPROCITY 21 CHAPTER 2. ARITHMETIC
FUNCTIONS 29 2.1 ARITHMETIC FUNCTIONS 29 2.2 AVERAGE ORDER 32 2.3
COUNTING THE NUMBER OF PRIMES 38 CHAPTER 3. ZETA AND L-FUNCTIONS 47 3.1
THE RIEMANN ZETA FUNCTION 47 3.2 ZEROS OF THE RIEMANN ZETA FUNCTION 54
3.3 DIRICHLET CHARACTERS AND .L-FUNCTIONS 69 CHAPTER 4. SOLUTIONS TO
DIOPHANTINE EQUATIONS 81 4.1 DIOPHANTINE EQUATIONS 81 4.2 ELLIPTIC
CURVES 85 4.3 HEIGHT FUNCTIONS AND DIOPHANTINE EQUATIONS 89 4.4 COUNTING
SOLUTIONS OF CONGRUENCES MODULO P 95 4.5 RESEARCH PROJECTS 105 PART 2.
CONTINUED FRACTIONS AND APPROXIMATIONS 107 CHAPTER 5. ALGEBRAIC AND
TRANSCENDENTAL NUMBERS 109 VIII CONTENTS 5.1 RUSSELL S PARADOX AND THE
BANACH-TARSKI PARADOX 109 5.2 DEFINITIONS 110 5.3 COUNTABLE AND
UNCOUNTABLE SETS 112 5.4 PROPERTIES OF E 118 5.5 EXPONENT (OR ORDER) OF
APPROXIMATION 124 5.6 LIOUVILLE S THEOREM 128 5.7 ROTH S THEOREM 132
CHAPTER 6. THE PROOF OF ROTH S THEOREM 137 6.1 LIOUVILLE S THEOREM AND
ROTH S THEOREM 137 6.2 EQUIVALENT FORMULATION OF ROTH S THEOREM 138 6.3
ROTH S MAIN LEMMA 142 6.4 PRELIMINARIES TO PROVING ROTH S LEMMA 147 6.5
PROOF OF ROTH S LEMMA 155 CHAPTER 7. INTRODUCTION TO CONTINUED FRACTIONS
158 7.1 DECIMAL EXPANSIONS 158 7.2 DEFINITION OF CONTINUED FRACTIONS 159
7.3 REPRESENTATION OF NUMBERS BY CONTINUED FRACTIONS 161 7.4 INFINITE
CONTINUED FRACTIONS 167 7.5 POSITIVE SIMPLE CONVERGENTS AND CONVERGENCE
169 7.6 PERIODIC CONTINUED FRACTIONS AND QUADRATIC IRRATIONALS 170 7.7
COMPUTING ALGEBRAIC NUMBERS CONTINUED FRACTIONS 177 7.8 FAMOUS CONTINUED
FRACTION EXPANSIONS 179 7.9 CONTINUED FRACTIONS AND APPROXIMATIONS 182
7.10 RESEARCH PROJECTS 186 PART 3. PROBABILISTIC METHODS AND
EQUIDISTRIBUTION 189 CHAPTER 8. INTRODUCTION TO PROBABILITY 191 8.1
PROBABILITIES OF DISCRETE EVENTS 192 8.2 STANDARD DISTRIBUTIONS 205 8.3
RANDOM SAMPLING 211 8.4 THE CENTRAL LIMIT THEOREM 213 CHAPTER 9.
APPLICATIONS OF PROBABILITY: BENFORD S LAW AND HYPOTHESIS TESTING 216
9.1 BENFORD S LAW 216 9.2 BENFORD S LAW AND EQUIDISTRIBUTED SEQUENCES
218 9.3 RECURRENCE RELATIONS AND BENFORD S LAW 219 9.4 RANDOM WALKS AND
BENFORD S LAW 221 9.5 STATISTICAL INFERENCE 225 9.6 SUMMARY 229 CHAPTER
10. DISTRIBUTION OF DIGITS OF CONTINUED FRACTIONS 231 10.1 SIMPLE
RESULTS ON DISTRIBUTION OF DIGITS 231 10.2 MEASURE OF A WITH SPECIFIED
DIGITS 235 CONTENTS IX 10.3 THE GAUSS-KUZMIN THEOREM 237 10.4
DEPENDENCIES OF DIGITS 244 10.5 GAUSS-KUZMIN EXPERIMENTS 248 10.6
RESEARCH PROJECTS 252 CHAPTER 11. INTRODUCTION TO FOURIER ANALYSIS 255
11.1 INNER PRODUCT OF FUNCTIONS 256 11.2 FOURIER SERIES 258 11.3
CONVERGENCE OF FOURIER SERIES 262 11.4 APPLICATIONS OF THE FOURIER
TRANSFORM 268 11.5 CENTRAL LIMIT THEOREM 273 11.6 ADVANCED TOPICS 276
CHAPTER 12. {N K A} AND POISSONIAN BEHAVIOR 278 12.1 DEFINITIONS AND
PROBLEMS 278 12.2 DENSENESSOF {N K A} 280 12.3 EQUIDISTRIBUTION OF {N K
A} 283 12.4 SPACING PRELIMINARIES 288 12.5 POINT MASSES AND INDUCED
PROBABILITY MEASURES 289 12.6 NEIGHBOR SPACINGS 290 12.7 POISSONIAN
BEHAVIOR 291 12.8 NEIGHBOR SPACINGS OF {N K A} 296 12.9 RESEARCH
PROJECTS 299 PART 4. THE CIRCLE METHOD 301 CHAPTER 13. INTRODUCTION TO
THE CIRCLE METHOD 303 13.1 ORIGINS 303 13.2 THE CIRCLE METHOD 309 13.3
GOLDBACH S CONJECTURE REVISITED 315 CHAPTER 14. CIRCLE METHOD:
HEURISTICS FOR GERMAIN PRIMES 326 14.1 GERMAIN PRIMES 326 14.2
PRELIMINARIES 328 14.3 THE FUNCTIONS FN {X) AND U(X) 331 14.4
APPROXIMATING FN (X) ON THE MAJOR ARCS 332 14.5 INTEGRALS OVER THE MAJOR
ARCS 338 14.6 MAJOR ARCS AND THE SINGULAR SERIES 342 14.7 NUMBER OF
GERMAIN PRIMES AND WEIGHTED SUMS 350 14.8 EXERCISES 353 14.9 RESEARCH
PROJECTS 354 PART 5. RANDOM MATRIX THEORY AND L-FUNCTIONS 357 CHAPTER
15. FROM NUCLEAR PHYSICS TO L-FUNCTIONS 359 15.1 HISTORICAL INTRODUCTION
359 15.2 EIGENVALUE PRELIMINARIES 364 X CONTENTS 15.3 SEMI-CIRCLE LAW
368 15.4 ADJACENT NEIGHBOR SPACINGS 374 15.5 THIN SUB-FAMILIES 377 15.6
NUMBER THEORY 383 15.7 SIMILARITIES BETWEEN RANDOM MATRIX THEORY AND
L-FUNCTIONS 389 15.8 SUGGESTIONS FOR FURTHER READING 390 CHAPTER 16.
RANDOM MATRIX THEORY: EIGENVALUE DENSITIES 391 16.1 SEMI-CIRCLE LAW 391
16.2 NON-SEMI-CIRCLE BEHAVIOR 398 16.3 SPARSE MATRICES 402 16.4 RESEARCH
PROJECTS 403 CHAPTER 17. RANDOM MATRIX THEORY: SPACINGS BETWEEN ADJACENT
EIGENVALUES 405 17.1 INTRODUCTION TO THE 2 X 2 GOE MODEL 405 17.2
DISTRIBUTION OF EIGENVALUES OF 2 X 2 GOE MODEL 409 17.3 GENERALIZATION
TO N X N GOE 414 17.4 CONJECTURES AND RESEARCH PROJECTS 418 CHAPTER 18.
THE EXPLICIT FORMULA AND DENSITY CONJECTURES 421 18.1 EXPLICIT FORMULA
422 18.2 DIRICHLET CHARACTERS FROM A PRIME CONDUCTOR 429 18.3 SUMMARY OF
CALCULATIONS 437 APPENDIX A. ANALYSIS REVIEW 439 A.I PROOFS BY INDUCTION
439 A.2 CALCULUS REVIEW 442 A.3 CONVERGENCE AND CONTINUITY 447 A.4
DIRICHLET S PIGEON-HOLE PRINCIPLE 448 A.5 MEASURES AND LENGTH 450 A. 6
INEQUALITIES 452 APPENDIX B. LINEAR ALGEBRA REVIEW 455 B.I DEFINITIONS
455 B.2 CHANGE OF BASIS 456 B.3 ORTHOGONAL AND UNITARY MATRICES 457 B.4
TRACE 458 B.5 SPECTRAL THEOREM FOR REAL SYMMETRIC MATRICES 459 APPENDIX
C. HINTS AND REMARKS ON THE EXERCISES 463 APPENDIX D. CONCLUDING REMARKS
475 BIBLIOGRAPHY 476 INDEX 497
|
| adam_txt |
AN INVITATION TO MODERN NUMBER THEORY STEVEN J. MILLER AND RAMIN
TAKLOO-BIGHASH PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD CONTENTS
FOREWORD XI PREFACE XIII NOTATION XIX PART 1. BASIC NUMBER THEORY 1
CHAPTER 1. MOD P ARITHMETIC, GROUP THEORY AND CRYPTOGRAPHY 3 1.1
CRYPTOGRAPHY 3 1.2 EFFICIENT ALGORITHMS 5 1.3 CLOCK ARITHMETIC:
ARITHMETIC MODULO N 14 1.4 GROUP THEORY 15 1.5 RSA REVISITED 20 1.6
EISENSTEIN'S PROOF OF QUADRATIC RECIPROCITY 21 CHAPTER 2. ARITHMETIC
FUNCTIONS 29 2.1 ARITHMETIC FUNCTIONS 29 2.2 AVERAGE ORDER 32 2.3
COUNTING THE NUMBER OF PRIMES 38 CHAPTER 3. ZETA AND L-FUNCTIONS 47 3.1
THE RIEMANN ZETA FUNCTION 47 3.2 ZEROS OF THE RIEMANN ZETA FUNCTION 54
3.3 DIRICHLET CHARACTERS AND .L-FUNCTIONS 69 CHAPTER 4. SOLUTIONS TO
DIOPHANTINE EQUATIONS 81 4.1 DIOPHANTINE EQUATIONS 81 4.2 ELLIPTIC
CURVES 85 4.3 HEIGHT FUNCTIONS AND DIOPHANTINE EQUATIONS 89 4.4 COUNTING
SOLUTIONS OF CONGRUENCES MODULO P 95 4.5 RESEARCH PROJECTS 105 PART 2.
CONTINUED FRACTIONS AND APPROXIMATIONS 107 CHAPTER 5. ALGEBRAIC AND
TRANSCENDENTAL NUMBERS 109 VIII CONTENTS 5.1 RUSSELL'S PARADOX AND THE
BANACH-TARSKI PARADOX 109 5.2 DEFINITIONS 110 5.3 COUNTABLE AND
UNCOUNTABLE SETS 112 5.4 PROPERTIES OF E 118 5.5 EXPONENT (OR ORDER) OF
APPROXIMATION 124 5.6 LIOUVILLE'S THEOREM 128 5.7 ROTH'S THEOREM 132
CHAPTER 6. THE PROOF OF ROTH'S THEOREM 137 6.1 LIOUVILLE'S THEOREM AND
ROTH'S THEOREM 137 6.2 EQUIVALENT FORMULATION OF ROTH'S THEOREM 138 6.3
ROTH'S MAIN LEMMA 142 6.4 PRELIMINARIES TO PROVING ROTH'S LEMMA 147 6.5
PROOF OF ROTH'S LEMMA 155 CHAPTER 7. INTRODUCTION TO CONTINUED FRACTIONS
158 7.1 DECIMAL EXPANSIONS 158 7.2 DEFINITION OF CONTINUED FRACTIONS 159
7.3 REPRESENTATION OF NUMBERS BY CONTINUED FRACTIONS 161 7.4 INFINITE
CONTINUED FRACTIONS 167 7.5 POSITIVE SIMPLE CONVERGENTS AND CONVERGENCE
169 7.6 PERIODIC CONTINUED FRACTIONS AND QUADRATIC IRRATIONALS 170 7.7
COMPUTING ALGEBRAIC NUMBERS'CONTINUED FRACTIONS 177 7.8 FAMOUS CONTINUED
FRACTION EXPANSIONS 179 7.9 CONTINUED FRACTIONS AND APPROXIMATIONS 182
7.10 RESEARCH PROJECTS 186 PART 3. PROBABILISTIC METHODS AND
EQUIDISTRIBUTION 189 CHAPTER 8. INTRODUCTION TO PROBABILITY 191 8.1
PROBABILITIES OF DISCRETE EVENTS 192 8.2 STANDARD DISTRIBUTIONS 205 8.3
RANDOM SAMPLING 211 8.4 THE CENTRAL LIMIT THEOREM 213 CHAPTER 9.
APPLICATIONS OF PROBABILITY: BENFORD'S LAW AND HYPOTHESIS TESTING 216
9.1 BENFORD'S LAW 216 9.2 BENFORD'S LAW AND EQUIDISTRIBUTED SEQUENCES
218 9.3 RECURRENCE RELATIONS AND BENFORD'S LAW 219 9.4 RANDOM WALKS AND
BENFORD'S LAW 221 9.5 STATISTICAL INFERENCE 225 9.6 SUMMARY 229 CHAPTER
10. DISTRIBUTION OF DIGITS OF CONTINUED FRACTIONS 231 10.1 SIMPLE
RESULTS ON DISTRIBUTION OF DIGITS 231 10.2 MEASURE OF A WITH SPECIFIED
DIGITS 235 CONTENTS IX 10.3 THE GAUSS-KUZMIN THEOREM 237 10.4
DEPENDENCIES OF DIGITS 244 10.5 GAUSS-KUZMIN EXPERIMENTS 248 10.6
RESEARCH PROJECTS 252 CHAPTER 11. INTRODUCTION TO FOURIER ANALYSIS 255
11.1 INNER PRODUCT OF FUNCTIONS 256 11.2 FOURIER SERIES 258 11.3
CONVERGENCE OF FOURIER SERIES 262 11.4 APPLICATIONS OF THE FOURIER
TRANSFORM 268 11.5 CENTRAL LIMIT THEOREM 273 11.6 ADVANCED TOPICS 276
CHAPTER 12. {N K A} AND POISSONIAN BEHAVIOR 278 12.1 DEFINITIONS AND
PROBLEMS 278 12.2 DENSENESSOF {N K A} 280 12.3 EQUIDISTRIBUTION OF {N K
A} 283 12.4 SPACING PRELIMINARIES 288 12.5 POINT MASSES AND INDUCED
PROBABILITY MEASURES 289 12.6 NEIGHBOR SPACINGS 290 12.7 POISSONIAN
BEHAVIOR 291 12.8 NEIGHBOR SPACINGS OF {N K A} 296 12.9 RESEARCH
PROJECTS 299 PART 4. THE CIRCLE METHOD 301 CHAPTER 13. INTRODUCTION TO
THE CIRCLE METHOD 303 13.1 ORIGINS 303 13.2 THE CIRCLE METHOD 309 13.3
GOLDBACH'S CONJECTURE REVISITED 315 CHAPTER 14. CIRCLE METHOD:
HEURISTICS FOR GERMAIN PRIMES 326 14.1 GERMAIN PRIMES 326 14.2
PRELIMINARIES 328 14.3 THE FUNCTIONS FN {X) AND U(X) 331 14.4
APPROXIMATING FN (X) ON THE MAJOR ARCS 332 14.5 INTEGRALS OVER THE MAJOR
ARCS 338 14.6 MAJOR ARCS AND THE SINGULAR SERIES 342 14.7 NUMBER OF
GERMAIN PRIMES AND WEIGHTED SUMS 350 14.8 EXERCISES 353 14.9 RESEARCH
PROJECTS 354 PART 5. RANDOM MATRIX THEORY AND L-FUNCTIONS 357 CHAPTER
15. FROM NUCLEAR PHYSICS TO L-FUNCTIONS 359 15.1 HISTORICAL INTRODUCTION
359 15.2 EIGENVALUE PRELIMINARIES 364 X CONTENTS 15.3 SEMI-CIRCLE LAW
368 15.4 ADJACENT NEIGHBOR SPACINGS 374 15.5 THIN SUB-FAMILIES 377 15.6
NUMBER THEORY 383 15.7 SIMILARITIES BETWEEN RANDOM MATRIX THEORY AND
L-FUNCTIONS 389 15.8 SUGGESTIONS FOR FURTHER READING 390 CHAPTER 16.
RANDOM MATRIX THEORY: EIGENVALUE DENSITIES 391 16.1 SEMI-CIRCLE LAW 391
16.2 NON-SEMI-CIRCLE BEHAVIOR 398 16.3 SPARSE MATRICES 402 16.4 RESEARCH
PROJECTS 403 CHAPTER 17. RANDOM MATRIX THEORY: SPACINGS BETWEEN ADJACENT
EIGENVALUES 405 17.1 INTRODUCTION TO THE 2 X 2 GOE MODEL 405 17.2
DISTRIBUTION OF EIGENVALUES OF 2 X 2 GOE MODEL 409 17.3 GENERALIZATION
TO N X N GOE 414 17.4 CONJECTURES AND RESEARCH PROJECTS 418 CHAPTER 18.
THE EXPLICIT FORMULA AND DENSITY CONJECTURES 421 18.1 EXPLICIT FORMULA
422 18.2 DIRICHLET CHARACTERS FROM A PRIME CONDUCTOR 429 18.3 SUMMARY OF
CALCULATIONS 437 APPENDIX A. ANALYSIS REVIEW 439 A.I PROOFS BY INDUCTION
439 A.2 CALCULUS REVIEW 442 A.3 CONVERGENCE AND CONTINUITY 447 A.4
DIRICHLET'S PIGEON-HOLE PRINCIPLE 448 A.5 MEASURES AND LENGTH 450 A. 6
INEQUALITIES 452 APPENDIX B. LINEAR ALGEBRA REVIEW 455 B.I DEFINITIONS
455 B.2 CHANGE OF BASIS 456 B.3 ORTHOGONAL AND UNITARY MATRICES 457 B.4
TRACE 458 B.5 SPECTRAL THEOREM FOR REAL SYMMETRIC MATRICES 459 APPENDIX
C. HINTS AND REMARKS ON THE EXERCISES 463 APPENDIX D. CONCLUDING REMARKS
475 BIBLIOGRAPHY 476 INDEX 497 |
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| author | Miller, Steven J. 1974- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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Showing how experiments are used to test conjectures and prove theorems, the book allows students to do original work on such problems, often using little more than calculus (though there are numerous remarks for those with deeper backgrounds). It shows students what number theory theorems are used for and what led to them and suggests problems for further research. Steven Miller and Ramin Takloo-Bighash introduce the problems and the computational skills required to numerically investigate them, providing background material (from probability to statistics to Fourier analysis) whenever necessary. They guide students through a variety of problems, ranging from basic number theory, cryptography, and Goldbach's Problem, to the algebraic structures of numbers and continued fractions, showing connections between these subjects and encouraging students to study them further. In addition, this is the first undergraduate book to explore Random Matrix Theory, which has recently become a powerful tool for predicting answers in number theory. Providing exercises, references to the background literature, and Web links to previous student research projects, An Invitation to Modern Number Theory can be used to teach a research seminar or a lecture class.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Nombres, Théorie des</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Number theory</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Zahlentheorie</subfield><subfield code="0">(DE-588)4067277-3</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Zahlentheorie</subfield><subfield code="0">(DE-588)4067277-3</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Takloo-Bighash, Ramin</subfield><subfield code="e">Sonstige</subfield><subfield code="4">oth</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">GBV Datenaustausch</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014788121&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-014788121</subfield></datafield></record></collection> |
| id | DE-604.BV021572335 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T14:38:50Z |
| indexdate | 2024-07-09T20:38:56Z |
| institution | BVB |
| isbn | 0691120609 9780691120607 |
| language | English |
| lccn | 2005052165 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014788121 |
| oclc_num | 61353135 |
| open_access_boolean | |
| owner | DE-20 DE-83 DE-11 |
| owner_facet | DE-20 DE-83 DE-11 |
| physical | XX, 503 S. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Princeton Univ. Press |
| record_format | marc |
| spelling | Miller, Steven J. 1974- Verfasser (DE-588)173874002 aut An invitation to modern number theory Steven J. Miller and Ramin Takloo-Bighash Princeton [u.a.] Princeton Univ. Press 2006 XX, 503 S. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index In a manner accessible to beginning undergraduates, An Invitation to Modern Number Theory introduces many of the central problems, conjectures, results, and techniques of the field, such as the Riemann Hypothesis, Roth's Theorem, the Circle Method, and Random Matrix Theory. Showing how experiments are used to test conjectures and prove theorems, the book allows students to do original work on such problems, often using little more than calculus (though there are numerous remarks for those with deeper backgrounds). It shows students what number theory theorems are used for and what led to them and suggests problems for further research. Steven Miller and Ramin Takloo-Bighash introduce the problems and the computational skills required to numerically investigate them, providing background material (from probability to statistics to Fourier analysis) whenever necessary. They guide students through a variety of problems, ranging from basic number theory, cryptography, and Goldbach's Problem, to the algebraic structures of numbers and continued fractions, showing connections between these subjects and encouraging students to study them further. In addition, this is the first undergraduate book to explore Random Matrix Theory, which has recently become a powerful tool for predicting answers in number theory. Providing exercises, references to the background literature, and Web links to previous student research projects, An Invitation to Modern Number Theory can be used to teach a research seminar or a lecture class. Nombres, Théorie des Number theory Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s DE-604 Takloo-Bighash, Ramin Sonstige oth GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014788121&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Miller, Steven J. 1974- An invitation to modern number theory Nombres, Théorie des Number theory Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4067277-3 |
| title | An invitation to modern number theory |
| title_auth | An invitation to modern number theory |
| title_exact_search | An invitation to modern number theory |
| title_exact_search_txtP | An invitation to modern number theory |
| title_full | An invitation to modern number theory Steven J. Miller and Ramin Takloo-Bighash |
| title_fullStr | An invitation to modern number theory Steven J. Miller and Ramin Takloo-Bighash |
| title_full_unstemmed | An invitation to modern number theory Steven J. Miller and Ramin Takloo-Bighash |
| title_short | An invitation to modern number theory |
| title_sort | an invitation to modern number theory |
| topic | Nombres, Théorie des Number theory Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Nombres, Théorie des Number theory Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014788121&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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