Textbook in PDF format Arising, as it does, from advanced lectures given in Bordeaux, Paris and Nancy over the past fifteen years (and for which an earlier English version is available from Cambridge University Press), this book is a revised, updated, and expanded version of a volume that appeared in 1990 in the Publications de l'Institut Élie Cartan. It was written with the purpose of providing young researchers with a self-contained introduction to the analytic methods of number theory, and their elders with a source of references for a number of fundamental questions. Such an undertaking necessarily involves choices. As these were made, they were generally taken on aesthetic grounds-not to forget the categorical imperatives imposed by ignorance. 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. Foreword Preface to the third edition Preface to the English translation Notation Part I. Elementary Methods Some tools from real analysis Abel summation The Euler-Maclaurin summation formula Exercises Prime numbers Introduction Chebyshev's estimates p-adic valuation of n! Mertens' first theorem Two new asymptotic formulae Merten's formula Another theorem of Chebyshev Notes Exercises Arithmetic Functions Definitions Examples Formal Dirichlet series The ring of arithmetic Functions The Mobius inversion formulae Von Mangoldt's Function Euler's totient Function Notes Exercises Average orders Introduction Dirichlet's problem and the hyperbola method The sum of divisors Function Euler's totient Function The Functions ω and Ω Mean value of the Möbius Function and Chebyshev's summatory Functions Squarefree integers Mean value of a multiplicative function with values in [0, 1] Notes Exercises Sieve methods The sieve of Eratosthenes Brun's combinatorial sieve Application to twin primes The large sieve–analytic form The large sieve–arithmetic form Applications of the large sieve Selberg's sieve Sums of two squares in an interval Notes Exercises Extremal orders Introduction and definitions The Function τ(n) The Functions ω(n) and Ω(n) Euler's Function φ(n) The Functions σk(n), k > 0 Notes Exercises The method of van der Corput Introduction and prerequisites Trigonometric integrals Trigonometric sums Application to Voronoï's theorem Equidistribution modulo 1 Notes Exercises Diophantine approximation From Dirichlet to Roth Best approximations, continued fractions Properties of the continued fraction expansion Continued fraction expansion of quadratic irrationals Notes Exercises Part II. Complex Analysis Methods The Euler Gamma Function Definitions The Weierstrass product formula The Beta Function Complex Stirling's formula Hankel's formula Exercises Generating Functions: Dirichlet series Convergent Dirichlet series Dirichlet series of multiplicative Functions Fundamental analytic properties of Dirichlet series Abscissa of convergence and mean value An arithmetic application: the core of an integer Order of magnitude in vertical strips Notes Exercises Summation formulae Perron formulae Applications: two convergence theorems The mean value formula Notes Exercises The Riemann zeta Function Introduction Analytic continuation Functional equation Approximations and bounds in the critical strip Initial localization of zeros Lemmas From complex analysis Global distribution of zeros Expansion as a Hadamard product Zero-free regions Bounds for ζ'/ζ, 1/ ζ and log ζ Notes Exercises The prime number theorem and the Riemann hypothesis The prime number theorem Minimal hypotheses The Riemann hypothesis Explicit formula for ψ(x) Notes Exercises The Selberg-Delange method Complex powers of ζ(s) The main result Proof of Theorem 5.2 A variant of the main theorem Notes Exercises Two arithmetic applications Integers having k prime factors The average distribution of divisors: the arcsine law Notes Exercises Tauberian Theorems Introduction. Abelian/Tauberian theorems duality Tauber's theorem The theorems of Hardy-Littlewood and Karamata The remainder term in Karamata's theorem Ikehara's theorem The Berry-Esseen inequality Holomorphy as a Tauberian condition Arithmetic Tauberian theorems Notes Exercises Primes in arithmetic progressions Introduction. Dirichlet characters L-series. The prime number theorem for arithmetic progressions Lower bounds for |L(s, χ)| when σ > 1. Proof of Theorem 8.16 The Functional equation for the Functions L(s, χ) Hadamard product formula and zero-free regions Explicit formulae for ψ(x; χ) Final form of the prime number theorem for arithmetic progressions Notes Exercises Part III. Probabilistic Methods Densities Definitions. Natural density Logarithmic density Analytic density Probabilistic number theory Notes Exercises Limiting distributions of arithmetic Functions Definition–distribution Functions Characteristic functions Notes Exercises Normal order Definition The Turán–Kubilius inequality Dual form of the Turán–Kubilius inequality The Hardy–Ramanujan theorem and other applications Effective mean value estimates for multiplicative Functions Normal structure of the sequence of prime factors of an integer Notes Exercises Distribution of additive Functions and mean values of multiplicative Functions The Erdös–Wintner theorem Delange's theorem Halasz's theorem The Erdös–Kac theorem Notes Exercises Friable integers. The saddle-point method Introduction. Rankin's method The geometric method Functional equations Dickman's Function Approximation to Ψ(x, y) by the saddle-point method Jacobsthal's Function and Rankin's theorem Notes Exercises Integers free of small prime factors Introduction Functional equations Buchstab's function Approximations to Ф(x, y) by the saddle-point method The Kubilius model Notes Exercises Bibliography Index