Introduction to Analytic and Probabilistic Number Theory (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 163)

This book provides a self contained, thorough introduction to the analytic and probabilistic methods of number theory. The prerequisites being reduced to classical contents of undergraduate courses, it offers to students and young researchers a systematic and consistent account on the subject. It is also a convenient tool for professional mathematicians, who may use it for basic references concerning many fundamental topics. Deliberately placing the methods before the results, the book will be of use beyond the particular material addressed directly. Each chapter is complemented with bibliographic notes, useful for descriptions of alternative viewpoints, and detailed exercises, often leading to research problems. This third edition of a text that has become classical offers a renewed and considerably enhanced content, being expanded by more than 50 percent. Important new developments are included, along with original points of view on many essential branches of arithmetic and an accurate perspective on up-to-date bibliography. The author has made important contributions to number theory and his mastery of the material is reflected in the exposition, which is lucid, elegant, and accurate. —Mathematical Reviews Cover Title page Dedication Contents Foreword Preface to the third edition Preface to the English translation Notation Part I. Elementary Methods Chapter 1.0. Some tools from real analysis §0.1. Abel summation §0.2. The Euler-Maclaurin summation formula Exercises Chapter 1.1. Prime numbers §1.1. Introduction §1.2. Chebyshev's estimates §1.3. p-adic valuation of n! §1.4. Mertens' first theorem §1.5. Two new asymptotic formulae §1.6. Merten's formula §1.7. Another theorem of Chebyshev Notes Exercises Chapter 1.2. Arithmetic functions §2.1. Definitions §2.2. Examples §2.3. Formal Dirichlet series §2.4. The ring of arithmetic functions §2.5. The Mobius inversion formulae §2.6. Von Mangoldt's function §2.7. Euler's totient function Notes Exercises Chapter 1.3. Average orders §3.1. Introduction §3.2. Dirichlet's problem and the hyperbola method §3.3. The sum of divisors function §3.4. Euler's totient function §3.5. The functions ? and ? §3.6. Mean value of the Mobius function and Chebyshev's summatory functions §3.7. Squarefree integers §3.8. Mean value of a multiplicative function with values in [0,1] Notes Exercises Chapter 1.4. Sieve methods §4.1. The sieve of Eratosthenes §4.2. Bran's combinatorial sieve §4.3. Application to twin primes §4.4. The large sieve—analytic form §4.5. The large sieve—arithmetic form §4.6. Applications of the large sieve §4.7. Selberg's sieve §4.8. Sums of two squares in an interval Notes Exercises Chapter 1.5. Extremal orders §5.1. Introduction and definitions §5.2. The function r(n) §5.3. The functions ?(?) and ?(?) §5.4. Euler's function ?{?) §5.5. The functions ??(?),? > 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 Vorono'i's theorem §6.5. Equidistribution modulo 1 Notes Exercises Chapter 1.7. Diophantine approximation §7.1. Prom 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 2.0. The Euler Gamma function §0.1. Definitions §0.2. The Weierstrass product formula §0.3. The Beta function §0.4. Complex Stirling's formula §0.5. HankePs formula Exercises Chapter 2.1 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 2.2. 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 C7C, 1/C and log Ñ 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 ?(?) Notes Exercises Chapter II. 5. The Selberg-Delange method §5.1. Complex powers of ?(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 ê 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,%)| when ? ^ 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 ?(?;?) §8.7. Final form of the prime number theorem for arithmetic progressions Notes Exercises Part III. Probabilistic Methods Chapter III.l. Densities §1.1. Definitions. Natural density §1.2. Logarithmic density §1.3. Analytic density §1.4. Probabilistic number theory Notes Exercises Chapter III.2. Limiting distributions of arithmetic functions §2.1. Definition—distribution functions §2.2. Characteristic functions Notes Exercises Chapter III.3. Normal order §3.1. Definition §3.2. The Turan-Kubilius inequality §3.3. Dual form of the Turan-Kubilius inequality §3.4. The Hardy-Ramanujan theorem and other applications §3.5. Effective mean value estimates for multiplicative functions §3.6. Normal structure of the sequence of prime factors of an integer Notes Exercises Chapter III.4. Distribution of additive functions and mean values of multiplicative functions §4.1. The Erdos-Wintner theorem §4.2. Delange's theorem §4.3. Halasz's theorem §4.4. The Erdos-Kac theorem 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 Ô(æ,ó) by the saddle-point method §5.6. JacobsthaFs 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 Ô(æ,ó) by the saddle-point method §6.5. The Kubilius model Notes Exercises Bibliography Index Solide initiation aux mthodes analytiques et probabilistes de l'arithmtique, ce livre constitue une rfrence indispensable. Ne s'appuyant que sur les connaissances traditionnellement enseignes en licence et master, il fournit en effet aux tudiants (notamment ceux qui prparent l'agrgation ou le CAPES de mathmatiques) et aux jeunes chercheurs une prsentation systmatique, cohrente et autonome du domaine. C'est galement un prcieux instrument de travail pour les mathmaticiens confirms sur nombre de questions fondamentales. Les mthodes plus que les rsultats sous-tendent le propos, de sorte que l'intrt de l'ouvrage dpasse largement le cadre strict de la thorie des nombres. Les chapitres sont par ailleurs complts de notes dtailles et de plus de 300 exercices de niveaux varis, certains dbouchant sur des problmes de recherche. Cette troisime dition d'un texte devenu classique, inclus dans la bibliothque de l'agrgation depuis de nombreuses annes, offre un contenu renouvel et considrablement enrichi. Elle comporte en particulier d'importants dveloppements indits, des points de vue originaux sur plusieurs branches essentielles de l'arithmtique, et une mise en perspective de la bibliographie la plus actuelle. Part I. Elementary Methods ; Prime Numbers ; Arithmetic Functions ; Average Orders ; Sieve Methods ; Extremal Orders ; The Method Of Van Der Corput ; Diophantine Approximation -- Part Ii. Complex Analysis Methods ; The Euler Gamma Function ; Generating Functions: Dirichlet Series ; Summation Formulae ; The Riemann Zeta Function ; The Prime Number Theorem And The Riemann Hypothesis ; The Selberg-delange Method ; Two Arithmetic Applications ; Tauberian Theorems ; Primes In Arithmetic Progressions -- Part Iii. Probabilistic Methods ; Densities ; Limiting Distributions Of Arithmetic Functions ; Normal Order ; Distribution Of Addictive Functions ; Friable Integers. The Saddle-point Method ; Integers Free Of Small Prime Factors. Gerald Tenenbaum ; Translated By Patrick D.f. Ion. Includes Bibliographical References (pages 591-615) And Index. English, Translated From The French.