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 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.