Introduction to Analytic and Probabilistic Number Theory (Cambridge Studies in Advanced Mathematics)

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Introduction to Analytic and Probabilistic Number Theory (Cambridge Studies in Advanced Mathematics)

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  • 製本 Hardcover:ハードカバー版
  • 言語 ENG,ENG
  • 商品コード 9780521412612
  • DDC分類 512.73

Full Description


This is a self-contained introduction to analytic methods in number theory, assuming on the part of the reader only what is typically learned in a standard undergraduate degree course. It offers to students and those beginning research a systematic and consistent account of the subject but will also be a convenient resource and reference for more experienced mathematicians. These aspects are aided by the inclusion at the end of each chapter of a section of bibliographic notes and detailed exercises. Professor Tenenbaum has emphasised methods rather than results, with the consequence that readers should be able to tackle more advanced material than is included here. Moreover, he has been able to cover developments on many new or unpublished topics such as: the Selberg-Delange method; a version of the Ikehara-Ingham Tauberian theorem; and a detailed exposition of the arithmetical use of the saddle-point method.

Table of Contents

Preface                                            xiii
Notation xv
Part I Elementary methods 1 (102)
Some tools from real analysis 3 (6)
Abel summation 3 (2)
The Euler-Maclaurin summation formula 5 (4)
Exercises 7 (2)
Prime numbers 9 (14)
Introduction 9 (1)
Chebyshev's estimates 10 (3)
p-adic valuation of n! 13 (1)
Mertens' first theorem 14 (1)
Two new asymptotic formulae 15 (2)
Mertens' formula 17 (2)
Another theorem of Chebyshev 19 (4)
Notes 20 (1)
Exercises 20 (3)
Arithmetic functions 23 (13)
Definitions 23 (1)
Examples 23 (2)
Formal Dirichlet series 25 (1)
The ring of arithmetic functions 26 (2)
The Mobius inversion formulae 28 (2)
Von Mangoldt's function 30 (2)
Euler's totient function 32 (4)
Notes 33 (1)
Exercises 34 (2)
Average orders 36 (20)
Introduction 36 (1)
Dirichlet's problem and the hyperbola 36 (3)
method
The sum of divisors function 39 (1)
Euler's totient function 39 (2)
The functions ω and Δ 41 (1)
Mean value of the Mobius function and the 42 (4)
summatory functions of Chebyshev
Squarefree integers 46 (2)
Mean value of a multiplicative function 48 (8)
with values in [0, 1]
Notes 50 (3)
Exercises 53 (3)
Sieve methods 56 (24)
The sieve of Eratosthenes 56 (1)
Brun's combinatorial sieve 57 (3)
Application to prime twins 60 (2)
The large sieve-analytic form 62 (6)
The large sieve-arithmetic form 68 (3)
Applications 71 (9)
Notes 74 (2)
Exercises 76 (4)
Extremal orders 80 (10)
Introduction and definitions 80 (1)
The function &tou;(n) 81 (2)
The functions ω(n) and Δ(n) 83 (1)
Euler's function φs;(n) 84 (1)
The functions φs;κ(n), κ 85 (5)
> 0
Notes 87 (1)
Exercises 87 (3)
The method of van der Corput 90 (13)
Introduction 90 (1)
Trigonometric integrals 91 (1)
Trigonometric sums 92 (4)
Application to the theorem of Voronoi 96 (7)
Notes 99 (1)
Exercises 100 (3)
Part II Methods of complex analysis 103 (164)
Generating functions: Dirichlet series 105 (25)
Convergent Dirichlet series 105 (1)
Dirichlet series of multiplicative 106 (1)
functions
Fundamental analytic properties of 107 (7)
Dirichlet series
Abscissa of convergence and mean value 114 (2)
An arithmetic application: the kernel of 116 (2)
an integer
Order of magnitude in vertical strips 118 (12)
Notes 122 (5)
Exercises 127 (3)
Summation formulae 130 (9)
Perron formulae 130 (4)
Application: a convergence theorem 134 (2)
The mean value formula 136 (3)
Notes 137 (1)
Exercises 138 (1)
The Riemann zeta function 139 (28)
Introduction 139 (1)
Analytic continuation 139 (3)
Functional equation 142 (1)
Approximations and bounds in the critical 143 (4)
strip
Initial localisation of zeros 147 (2)
Lemmas from complex analysis 149 (2)
Global distribution of zeros 151 (4)
Expansion as a Hadamard product 155 (2)
Zero-free regions 157 (1)
Bounds for ζ/ζ, 1/ζ and 158 (9)
logζ
Notes 160 (2)
Exercises 162 (5)
The prime number theorem and the Riemann 167 (13)
hypothesis
The prime number theorem 167 (1)
Minimal hypotheses 168 (2)
The Riemann hypothesis 170 (10)
Notes 174 (3)
Exercises 177 (3)
The Selberg-Delange method 180 (20)
Complex powers of ζ(s) 180 (3)
Hankel's formula 183 (1)
The main result 184 (3)
Proof of Theorem 3 187 (4)
A variant of the main theorem 191 (9)
Notes 195 (2)
Exercises 197 (3)
Two arithmetic applications 200 (17)
Integers having k prime factors 200 (7)
The average distribution of divisors: the 207 (10)
arcsine law
Notes 212 (2)
Exercises 214 (3)
Tauberian theorems 217 (31)
Introduction: Abelian/Tauberian theorems 217 (3)
duality
Tauber's theorem 220 (2)
The theorems of Hardy-Littlewood and 222 (5)
Karamata
The remainder term in Karamata's theorem 227 (7)
Ikehara's theorem 234 (6)
The Berry-Esseen inequality 240 (8)
Notes 242 (2)
Exercises 244 (4)
Prime numbers in arithmetic progressions 248 (19)
Introduction: Dirichlet characters 248 (4)
L-series. The prime number theorem for 252 (4)
arithmetic progressions
Lower bounds for /L(s, x)/ when σ 256 (11)
≥ 1. Proof of Theorem 4
Notes 262 (2)
Exercises 264 (3)
Part III Probabilistic methods 267 (157)
Densities 269 (12)
Definitions. Natural density 269 (3)
Logarithmic density 272 (1)
Analytic density 273 (2)
Probabilistic number theory 275 (6)
Notes 275 (1)
Exercises 276 (5)
Limiting distribution of arithmetic 281 (18)
functions
Definition--distribution functions 281 (4)
Characteristic functions 285 (14)
Notes 288 (7)
Exercises 295 (4)
Normal order 299 (26)
Definition 299 (1)
The Turan-Kubilius inequality 300 (4)
Dual form of the Turan-Kubilius inequality 304 (1)
The Hardy-Ramanujan theorem and other 305 (3)
applications
Effective mean value estimates for 308 (3)
multiplicative functions
Normal structure of the set of prime 311 (14)
factors of an integer
Notes 313 (6)
Exercises 319 (6)
Distribution of additive functions and mean 325 (33)
values of multiplicative functions
The Erdos-Wintner theorem 325 (6)
Delange's theorem 331 (4)
Halasz' theorem 335 (12)
The Erdos-Kac theorem 347 (11)
Notes 350 (3)
Exercises 353 (5)
Integers free of large prime factors. The 358 (37)
saddle-point method
Introduction. Rankin's method 358 (5)
The geometric method 363 (2)
Functional equations 365 (5)
Dickman's function 370 (7)
Approximations to ψ (x, y) by the 377 (18)
saddle-point method
Notes 387 (4)
Exercises 391 (4)
Integers free of small prime factors 395 (29)
Introduction 395 (3)
Functional equations 398 (5)
Buchstab's function 403 (5)
Approximations to φ (x, y) by the 408 (16)
saddle-point method
Notes 418 (2)
Exercises 420 (4)
Bibliography 424 (19)
Index 443