Introduction to analytic and probabilistic number theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
1995
|
| Edition: | 1. publ. in English |
| Series: | Cambridge studies in advanced mathematics
46 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 448 S. |
| ISBN: | 0521412617 |
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| 240 | 1 | 0 | |a Introduction à la théorie analytique et probabiliste des nombres |
| 245 | 1 | 0 | |a Introduction to analytic and probabilistic number theory |c Gérald Tenenbaum |
| 250 | |a 1. publ. in English | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 1995 | |
| 300 | |a XIV, 448 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Cambridge studies in advanced mathematics |v 46 | |
| 650 | 7 | |a Analytische getaltheorie |2 gtt | |
| 650 | 7 | |a Getaltheorie |2 gtt | |
| 650 | 7 | |a Nombres, Théorie des |2 ram | |
| 650 | 7 | |a Teoria analítica dos números |2 larpcal | |
| 650 | 7 | |a Teoria dos números |2 larpcal | |
| 650 | 7 | |a Waarschijnlijkheidstheorie |2 gtt | |
| 650 | 4 | |a Number theory | |
| 650 | 4 | |a Probabilistic number theory | |
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Record in the Search Index
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Contents
Preface xiii
Notation xv
Part I Elementary methods 1
Chapter 1.0 Some tools from real analysis 3
§ 0.1 Abel summation 3
§ 0.2 The Euler Maclaurin summation formula 5
Exercises 7
Chapter I.I Prime numbers 9
§ 1.1 Introduction 9
§ 1.2 Chebyshev's estimates 10
§ 1.3 p adic valuation of n! 13
§ 1.4 Mertens' first theorem 14
§ 1.5 Two new asymptotic formulae 15
§ 1.6 Mertens' formula 17
§ 1.7 Another theorem of Chebyshev 19
Notes 20
Exercises 20
Chapter 1.2 Arithmetic functions 23
§ 2.1 Definitions 23
§ 2.2 Examples 23
§ 2.3 Formal Dirichlet series 25
§ 2.4 The ring of arithmetic functions 26
§ 2.5 The Mobius inversion formulae 28
§ 2.6 Von Mangoldt's function 30
§ 2.7 Euler's totient function 32
Notes 33
Exercises 34
Chapter 1.3 Average orders 36
§ 3.1 Introduction 36
§ 3.2 Dirichlet's problem and the hyperbola method 36
§ 3.3 The sum of divisors function 39
§ 3.4 Euler's totient function 39
§ 3.5 The functions u and f2 41
§ 3.6 Mean value of the Mobius function and the summatory
functions of Chebyshev 42
§ 3.7 Squarefree integers 46
viii Contents
§ 3.8 Mean value of a multiplicative function with values in [0,1] 48
Notes 50
Exercises 53
Chapter 1.4 Sieve methods 56
§ 4.1 The sieve of Eratosthenes 56
§ 4.2 Brun's combinatorial sieve 57
§ 4.3 Application to prime twins 60
§ 4.4 The large sieve analytic form 62
§ 4.5 The large sieve arithmetic form 68
§ 4.6 Applications 71
Notes 74
Exercises 76
Chapter 1.5 Extremal orders 80
§ 5.1 Introduction and definitions 80
§ 5.2 The function r{n) 81
§ 5.3 The functions ui(n) and J7(n) 83
§ 5.4 Euler's function /?(n) 84
§ 5.5 The functions aK(n),k 0 85
Notes 87
Exercises 87
Chapter 1.6 The method of van der Corput 90
§ 6.1 Introduction 90
§ 6.2 Trigonometric integrals 91
§ 6.3 Trigonometric sums 92
§ 6.4 Application to the theorem of Vorono'i 96
Notes 99
Exercises 100
Part II Methods of complex analysis 103
Chapter II. 1 Generating functions: Dirichlet series 105
§ 1.1 Convergent Dirichlet series 105
§ 1.2 Dirichlet series of multiplicative functions 106
§ 1.3 Fundamental analytic properties of Dirichlet series 107
§ 1.4 Abscissa of convergence and mean value 114
§ 1.5 An arithmetic application: the kernel of an integer 116
§ 1.6 Order of magnitude in vertical strips 118
Notes 122
Exercises 127
Contents ix
Chapter II.2 Summation formulae 130
§ 2.1 Perron formulae 130
§ 2.2 Application : a convergence theorem 134
§ 2.3 The mean value formula 136
Notes 137
Exercises 138
Chapter II.3 The Riemann zeta function 139
§ 3.1 Introduction 139
§ 3.2 Analytic continuation 139
§ 3.3 Functional equation 142
§ 3.4 Approximations and bounds in the critical strip 143
§ 3.5 Initial localisation of zeros 147
§ 3.6 Lemmas from complex analysis 149
§ 3.7 Global distribution of zeros 151
§ 3.8 Expansion as a Hadamard product 155
§ 3.9 Zero free regions 157
§ 3.10 Bounds for C'/C, 1/C and logC 158
Notes 160
Exercises 162
Chapter II.4 The prime number theorem
and the Riemann hypothesis 167
§ 4.1 The prime number theorem 167
§ 4.2 Minimal hypotheses 168
§ 4.3 The Riemann hypothesis 170
Notes 174
Exercises 177
Chapter II.5 The Selberg Delange method 180
§ 5.1 Complex powers of C(«) 180
§ 5.2 Hankel's formula 183
§ 5.3 The main result 184
§ 5.4 Proof of Theorem 3 187
§ 5.5 A variant of the main theorem 191
Notes 195
Exercises 197
Chapter II.6 Two arithmetic applications 200
§ 6.1 Integers having k prime factors 200
§ 6.2 The average distribution of divisors: the arcsine law 207
Notes 212
Exercises 214
X Contents
Chapter II.7 Tauberian theorems 217
§ 7.1 Introduction: Abelian/Tauberian theorems duality 217
§ 7.2 Tauber's theorem 220
§ 7.3 The theorems of Hardy Littlewood and Karamata 222
§ 7.4 The remainder term in Karamata's theorem 227
§ 7.5 Ikehara's theorem 234
§ 7.6 The Berry Esseen inequality 240
Notes 242
Exercises 244
Chapter II.8 Prime numbers in arithmetic progressions 248
§ 8.1 Introduction: Dirichlet characters 248
§ 8.2 L series. The prime number theorem for arithmetic
progressions 252
§ 8.3 Lower bounds for \L(s, x)\ when a 1. Proof of
Theorem 4 256
Notes 262
Exercises 264
Part III Probabilistic methods 267
Chapter III.l Densities 269
§ 1.1 Definitions. Natural density 269
§ 1.2 Logarithmic density , 272
§ 1.3 Analytic density 273
§ 1.4 Probabilistic number theory 275
Notes 275
Exercises 276
Chapter III.2 Limiting distribution of arithmetic functions . 281
§ 2.1 Definition distribution functions 281
§ 2.2 Characteristic functions 285
Notes 288
Exercises 295
Chapter III.3 Normal order 299
§ 3.1 Definition 299
§ 3.2 The Turan Kubilius inequality 300
§ 3.3 Dual form of the Turan Kubilius inequality 304
§ 3.4 The Hardy Ramanujan theorem and other applications . 305
§ 3.5 Effective mean value estimates for multiplicative functions 308
§ 3.6 Normal structure of the set of prime factors of an integer 311
Notes 313
Exercises 319
Contents xi
Chapter III.4 Distribution of additive functions and
mean values of multiplicative functions 325
§ 4.1 The Erdos Wintner theorem 325
§ 4.2 Delange's theorem 331
§ 4.3 Halasz' theorem 335
§ 4.4 The Erdos Kac theorem 347
Notes 350
Exercises 353
Chapter III.5 Integers free of large prime factors.
The saddle point method 358
§ 5.1 Introduction. Rankin's method 358
§ 5.2 The geometric method 363
§ 5.3 Functional equations 365
§ 5.4 Dickman's function 370
§ 5.5 Approximations to ^!(x,y) by the saddle point method . 377
Notes 387
Exercises 391
Chapter III.6 Integers free of small prime factors 395
§ 6.1 Introduction 395
§ 6.2 Functional equations 398
§ 6.3 Buchstab's function 403
§ 6.4 Approximations to £(z, y) by the saddle point method . 408
Notes 418
Exercises 420
Bibliography 424
Index 443 |
| any_adam_object | 1 |
| author | Tenenbaum, Gérald 1952- |
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| author_facet | Tenenbaum, Gérald 1952- |
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| dewey-full | 512/.73 |
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| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.73 |
| dewey-search | 512/.73 |
| dewey-sort | 3512 273 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. in English |
| format | Book |
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| isbn | 0521412617 |
| language | English French |
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| series2 | Cambridge studies in advanced mathematics |
| spelling | Tenenbaum, Gérald 1952- Verfasser (DE-588)1066273146 aut Introduction à la théorie analytique et probabiliste des nombres Introduction to analytic and probabilistic number theory Gérald Tenenbaum 1. publ. in English Cambridge [u.a.] Cambridge Univ. Press 1995 XIV, 448 S. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 46 Analytische getaltheorie gtt Getaltheorie gtt Nombres, Théorie des ram Teoria analítica dos números larpcal Teoria dos números larpcal Waarschijnlijkheidstheorie gtt Number theory Probabilistic number theory Analytische Zahlentheorie (DE-588)4001870-2 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Analytische Zahlentheorie (DE-588)4001870-2 s DE-188 Zahlentheorie (DE-588)4067277-3 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s 1\p DE-604 Cambridge studies in advanced mathematics 46 (DE-604)BV000003678 46 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007016811&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Tenenbaum, Gérald 1952- Introduction to analytic and probabilistic number theory Cambridge studies in advanced mathematics Analytische getaltheorie gtt Getaltheorie gtt Nombres, Théorie des ram Teoria analítica dos números larpcal Teoria dos números larpcal Waarschijnlijkheidstheorie gtt Number theory Probabilistic number theory Analytische Zahlentheorie (DE-588)4001870-2 gnd Zahlentheorie (DE-588)4067277-3 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| subject_GND | (DE-588)4001870-2 (DE-588)4067277-3 (DE-588)4079013-7 (DE-588)4064324-4 |
| title | Introduction to analytic and probabilistic number theory |
| title_alt | Introduction à la théorie analytique et probabiliste des nombres |
| title_auth | Introduction to analytic and probabilistic number theory |
| title_exact_search | Introduction to analytic and probabilistic number theory |
| title_full | Introduction to analytic and probabilistic number theory Gérald Tenenbaum |
| title_fullStr | Introduction to analytic and probabilistic number theory Gérald Tenenbaum |
| title_full_unstemmed | Introduction to analytic and probabilistic number theory Gérald Tenenbaum |
| title_short | Introduction to analytic and probabilistic number theory |
| title_sort | introduction to analytic and probabilistic number theory |
| topic | Analytische getaltheorie gtt Getaltheorie gtt Nombres, Théorie des ram Teoria analítica dos números larpcal Teoria dos números larpcal Waarschijnlijkheidstheorie gtt Number theory Probabilistic number theory Analytische Zahlentheorie (DE-588)4001870-2 gnd Zahlentheorie (DE-588)4067277-3 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| topic_facet | Analytische getaltheorie Getaltheorie Nombres, Théorie des Teoria analítica dos números Teoria dos números Waarschijnlijkheidstheorie Number theory Probabilistic number theory Analytische Zahlentheorie Zahlentheorie Wahrscheinlichkeitstheorie Wahrscheinlichkeitsrechnung |
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