Introduction to Analytic and Probabilistic Number Theory: Third Edition
About this Title
Gérald Tenenbaum, Institut Élie Cartan, Vandoeuvre-lès Nancy, France. Translated by Dr Patrick D F Ion
Publication: Graduate Studies in Mathematics
Publication Year: 2015; Volume 163
ISBNs: 978-0-8218-9854-3 (print); 978-1-4704-2223-3 (online)
MathSciNet review: MR3363366
MSC: Primary 11-02; Secondary 11Kxx, 11Mxx, 11Nxx
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.
Graduate students and research mathematicians interested in number theory, analysis, and probability.
Table of Contents
Part I. Elementary methods
- Chapter I.0. Some tools from real analysis
- Chapter I.1. Prime numbers
- Chapter I.2. Arithmetic functions
- Chapter I.3. Average orders
- Chapter I.4. Sieve methods
- Chapter I.5. Extremal orders
- Chapter I.6. The method of van der Corput
- Chapter I.7. Diophantine approximation
Part II. Complex analysis methods
- Chapter II.0. The Euler gamma function
- Chapter II.1. Generating functions: Dirichlet series
- Chapter II.2. Summation formulae
- Chapter II.3. The Riemann zeta function
- Chapter II.4. The prime number theorem and the Riemann hypothesis
- Chapter II.5. The Selberg-Delange method
- Chapter II.6. Two arithmetic applications
- Chapter II.7. Tauberian theorems
- Chapter II.8. Primes in arithmetic progressions
Part III. Probabilistic methods
- Chapter III.1. Densities
- Chapter III.2. Limiting distributions of arithmetic functions
- Chapter III.3. Normal order
- Chapter III.4. Distribution of additive functions and mean values of multiplicative functions
- Chapter III.5. Friable integers. The saddle-point method
- Chapter III.6. Integers free of small factors