Introduction to analytic and probabilistic number theory:
Gespeichert in:
| 1. Verfasser: | |
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| Weitere Verfasser: | |
| Format: | Buch |
| Sprache: | English French |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2015]
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| Ausgabe: | Third edition |
| Schriftenreihe: | Graduate Studies in Mathematics
volume 163 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xxiv, 629 Seiten |
| ISBN: | 9780821898543 |
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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 ; translated by Patrick D.F. Ion |
| 250 | |a Third edition | ||
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2015] | |
| 264 | 4 | |c © 2015 | |
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| 650 | 4 | |a Probabilistic number theory | |
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Datensatz im Suchindex
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Titel: Introduction to analytic and probabilistic number theory
Autor: Tenenbaum, Gérald
Jahr: 2015
Contents Foreword xv Preface to the third edition xix Preface to the English translation xxi Notation xxiii Part I. Elementary Methods Chapter 1.0. Some tools from real analysis 3 §0.1. Abel summation 3 §0.2. The Euler-Maclaurin summation formula 5 Exercises 8 Chapter 1.1. Prime numbers 11 §1.1. Introduction 11 §1.2. Chebyshev’s estimates 13 §1.3. p-adic valuation of n! 15 §1.4. Mertens’ first theorem 16 §1.5. Two new asymptotic formulae 17 §1.6. Merten’s formula 19 §1.7. Another theorem of Chebyshev 20 Notes 22 Exercises 23 vii
Contents viii Chapter 1.2. Arithmetic functions 29 §2.1. Definitions 29 §2.2. Examples 30 §2.3. Formal Dirichlet series 31 §2.4. The ring of arithmetic functions 32 §2.5. The Möbius inversion formulae 34 §2.6. Von Mangoldt’s function 36 §2.7. Euler’s totient function 37 Notes 39 Exercises 40 Chapter 1.3. Average orders ? 43 §3.1. Introduction 43 §3.2. Dirichlet’s problem and the hyperbola method 44 §3.3. The sum of divisors function 46 §3.4. Euler’s totient function 46 §3.5. The functions to and Q 48 §3.6. Mean value of the Möbius function and Chebyshev’s summatory functions 49 §3.7. Squarefree integers 52 §3.8. Mean value of a multiplicative function with values in [0,1] 54 Notes 57 Exercises 59 Chapter 1.4. Sieve methods 67 §4.1. The sieve of Eratosthenes 67 §4.2. Brun’s combinatorial sieve 68 §4.3. Application to twin primes 71 §4.4. The large sieve—analytic form 73 §4.5. The large sieve—arithmetic form 79 §4.6. Applications of the large sieve 82 §4.7. Selberg’s sieve 84 §4.8. Sums of two squares in an interval 96 Notes 100 Exercises 105
Contents IX Chapter 1.5. Extremal orders §5.1. Introduction and definitions §5.2. The function r(n) §5.3. The functions w(n) and fi(n) §5.4. Euler’s function p(n) §5.5. The functions cr K (n), k 0 Notes Exercises Chapter 1.6. The method of van der Corput §6.1. Introduction and prerequisites §6.2. Trigonometric integrals §6.3. Trigonometric sums §6.4. Application to Voronoi’s theorem §6.5. Equidistribution modulo 1 Notes Exercises Chapter 1.7. Diophantine approximation §7.1. From Dirichlet to Roth §7.2. Best approximations, continued fractions §7.3. Properties of the continued fraction expansion §7.4. Continued fraction expansion of quadratic irrationals Notes Exercises Part II. Complex Analysis Methods Chapter II.0. The Euler Gamma function §0.1. Def ini tions §0.2. The Weierstrass product formula §0.3. The Beta function §0.4. Complex Stirling’s formula §0.5. Hankel’s formula Exercises 111 111 112 114 115 116 118 119 123 123 124 125 131 134 137 140 145 145 147 153 156 159 160 169 169 171 172 175 179 181
X Contents Chapter II.l. Generating functions: Dirichlet series §1.1. Convergent Dirichlet series §1.2. Dirichlet series of multiplicative functions §1.3. Fundamental analytic properties of Dirichlet series §1.4. Abscissa of convergence and mean value §1.5. An arithmetic application: the core of an integer §1.6. Order of magnitude in vertical strips Notes Exercises Chapter II.2. Summation formulae §2.1. Perron formulae §2.2. Applications: two convergence theorems §2.3. The mean value formula Notes Exercises Chapter II.3. The Riemann zeta function §3.1. Introduction §3.2. Analytic continuation §3.3. Functional equation §3.4. Approximations and bounds in the critical strip §3.5. Initial localization of zeros §3.6. Lemmas from complex analysis §3.7. Global distribution of zeros §3.8. Expansion as a Hadamard product §3.9. Zero-free regions §3.10. Bounds for C'/C L/C and logC Notes Exercises Chapter II.4. The prime number theorem and the Riemann hypothesis §4.1. The prime number theorem §4.2. Minimal hypotheses §4.3. The Riemann hypothesis §4.4. Explicit formula for ip(x) 187 187 188 189 196 198 200 204 211 217 217 223 225 227 228 231 231 232 234 235 238 240 242 245 247 248 251 254 261 261 262 264 268
Contents xi Notes Exercises Chapter II.5. The Selberg Delange method §5.1. Complex powers of Ç(s) §5.2. The main result §5.3. Proof of Theorem 5.2 §5.4. A variant of the main theorem Notes Exercises Chapter II.6. Two arithmetic applications §6.1. Integers having k prime factors §6.2. The average distribution of divisors: the arcsine law Notes Exercises Chapter II. 7. Tauberian Theorems §7.1. Introduction. Abelian/Tauberian theorems duality §7.2. Tauber’s theorem §7.3. The theorems of Hardy Littlewood and Karamata §7.4. The remainder term in Karamata’s theorem §7.5. Ikehara’s theorem §7.6. The Berry-Esseen inequality §7.7. Holomorphy as a Tauberian condition §7.8. Arithmetic Tauberian theorems Notes Exercises Chapter II.8. Primes in arithmetic progressions §8.1. Introduction. Dirichlet characters §8.2. L-series. The prime number theorem for arithmetic progressions §8.3. Lower bounds for |L(s,x)[ when a ^ 1. Proof of Theorem 8.16 §8.4. The functional equation for the functions L(s, x) §8.5. Hadamard product formula and zero-free regions §8.6. Explicit formulae for ip(x: x) 272 275 277 277 280 282 286 290 292 299 299 305 311 314 317 317 320 322 327 334 340 341 345 349 354 359 359 369 376 382 385 390
Contents xii §8.7. Final form of the prime number theorem for arithmetic progressions Notes Exercises 395 401 404 Part III. Probabilistic Methods Chapter III.l. Densities 413 §1.1. Definitions. Natural density 413 §1.2. Logarithmic density 416 §1.3. Analytic density 417 §1.4. Probabilistic number theory 419 Notes 420 Exercises 421 Chapter III.2. Limiting distributions of arithmetic functions 425 §2.1. Definition—distribution functions 425 §2.2. Characteristic functions 429 Notes 433 Exercises 440 Chapter III.3. Normal order 445 §3.1. Definition 445 §3.2. The Turân-Kubilius inequality 446 §3.3. Dual form of the Turân-Kubilius inequality 452 §3.4. The Hardy-Ramanujan theorem and other applications 453 §3.5. Effective mean value estimates for multiplicative functions 456 §3.6. Normal structure of the sequence of prime factors of an integer 459 Notes 461 Exercises 467 Chapter III.4. Distribution of additive functions and mean values of multiplicative functions 475 §4.1. The Erdos-Wintner theorem 475 §4.2. Delange’s theorem 481 §4.3. Halasz’s theorem 485 §4.4. The Erdos-Kac theorem 498
Contents xiii Notes Exercises Chapter III.5. Friable integers. The saddle-point method §5.1. Introduction. Rankin’s method §5.2. The geometric method §5.3. Functional equations §5.4. Dickman’s function §5.5. Approximation to T(a;, y) by the saddle-point method §5.6. Jacobsthal’s function and Rankin’s theorem Notes Exercises Chapter III.6. Integers free of small prime factors §6.1. Introduction §6.2. Functional equations §6.3. Buchstab’s function §6.4. Approximations to $(x,y) by the saddle-point method §6.5. The Kubilius model Notes Exercises Bibliography Index 501 505 511 511 516 518 523 530 539 543 552 557 557 560 564 569 579 583 588 591 617 |
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| 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 ; translated by Patrick D.F. Ion Third edition Providence, Rhode Island American Mathematical Society [2015] © 2015 xxiv, 629 Seiten txt rdacontent n rdamedia nc rdacarrier Graduate Studies in Mathematics volume 163 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 Ion, Patrick D. F. (DE-588)1166912256 trl Erscheint auch als Online-Ausgabe 978-1-4704-2223-3 Graduate Studies in Mathematics volume 163 (DE-604)BV009739289 163 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028200704&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 Graduate Studies in 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 ; translated by Patrick D.F. Ion |
| title_fullStr | Introduction to analytic and probabilistic number theory Gérald Tenenbaum ; translated by Patrick D.F. Ion |
| title_full_unstemmed | Introduction to analytic and probabilistic number theory Gérald Tenenbaum ; translated by Patrick D.F. Ion |
| 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 |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028200704&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT tenenbaumgerald introductionalatheorieanalytiqueetprobabilistedesnombres AT ionpatrickdf introductionalatheorieanalytiqueetprobabilistedesnombres AT tenenbaumgerald introductiontoanalyticandprobabilisticnumbertheory AT ionpatrickdf introductiontoanalyticandprobabilisticnumbertheory |