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Analytic number theory. Exploring the anatomy of integers. (English) Zbl 1247.11001

Graduate Studies in Mathematics 134. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-7577-3/hbk). xviii, 414 p. (2012).
The aim of the book under review is to provide a special introduction to the classical, highly venerable field of analytic number theory. As the authors point out in the preface to the book, the choice of the individual subtitle “Exploring the Anatomy of Integers” was coined at a CRM workshop held at the Université de Montréal, Canada, which the two authors, along with A. Granville, had organized in 2006. For that workshop as well as for the present introductory text, the terminology “anatomy of integers” was to describe the area of multiplicative number theory that relates to the size and distribution of the prime factors of positive integers and of various families of integers of particular interest. Thus the choice of subjects treated in this book includes prime numbers and their properties, the distribution of prime numbers, D. J. Newman’s analytic proof of the famous Prime Number Theorem, various arithmetic functions related to the multiplicative structure of the integers, sieve methods, primes in arithmetic progressions and their applications, the more recent “abc conjecture”, and other celebrated conjectures in analytic number theory.
As to the precise contents, the book consists of sixteen chapters, in which the authors assort a panoramic collection of fascinating topics from those parts of number theory where the use of analysis (real or complex) is of crucial importance.
Chapter 1 presents some basic notions and tools, mainly integral and summation formulas, fundamental inequalities, and some combinatorial results. Chapter 2 discusses prime numbers and their elementary properties, including estimates for the prime number function \(\pi(x)\) and some conjectures on the distribution of primes. Chapter 3 is devoted to Riemann’s zeta function and its analytic properties, while Chapter 4 deals with some classical arithmetic functions related to the Prime number Theorem. This sets the stage for the proof of the Prime Number Theorem in the subsequent chapter, chiefly by deriving various useful classical estimates for those basic arithmetic functions. Chapter 5 presents then an analytic proof of the Prime Number Theorem, basically by explaining D. J. Newman’s beautiful and relatively simple approach from 1980.
Chapter 6 introduces Dirichlet series and further arithmetic functions, touches upon average orders of additive functions, and gives a proof of A. Wintner’s theorem (1944) in this context. The local behavior of arithmetic functions is the main topic of Chapter 7 where also some still open problems of elementary nature are briefly commented on. Chapter 8 turns to the Euler phi-function and its arithmetic properties, before the prime factors of an integer are analyzed more closely in Chapter 9. The reader gets here acquainted with R. A. Rankin’s method, the application of pseudo-primes, the asymptotic behavior of Dickman’s function, and the properties of (consecutive) smooth numbers. Chapter 10 continues the study of arithmetic functions by briefly discussing the Hardy-Ramanujan inequality and an earlier, related theorem by E. Landau (1900 ).
The deep “abc conjecture” by D. W. Masser and J. Oesterlé (1985) is depicted in Chapter 11, together with the generalized Fermat equation, consecutive powerful numbers, the Erdős-Woods conjecture, and other related conjectures. Sieve methods are the principal theme of Chapter 12, where also twin primes, the Goldbach conjecture, the Brun-Titchmarsh theorem, the Schnirelman theorem, the Selberg sieve, the large sieve, quasi-squares, and other allied topics are encountered. Chapter 13 deals with prime numbers in arithmetic progressions, with the Primitive Divisor Theorem as a first main result in this context. Characters, primitive roots, \(L\)-series, and the completion of the proof of Dirichlet’s theorem on primes in arithmetic progressions are then treated in Chapter 14, and selected applications of this theory are outlined in the following Chapter 15. These applications concern Diophantine equations, the distribution of primes \(p\) with \(p-1\) squarefree, and a probabilistic result on asymptotic densities. Chapter 16, the final chapter of the book, is devoted to the study of the index of composition of an integer à la J.-M. De Koninck and N. Doyon [Monatsh. Math. 139, No. 2, 151–167 (2003; Zbl 1039.11059)], including elementary results, the description of both the local behavior and the distribution function of the crucial \(\lambda\)-function, and some related probabilistic results.
There is an appendix titled “Basic Complex Analysis Theory”, where the needed tools from complex analysis are briefly assembled. Apart from the lucid and detailed presentation of the material, this book stands out by another particular feature. Namely, in order to help the reader better comprehend the various themes, concepts, methods, techniques, and results presented in the course of the book, the authors have listed a collection of problems at the end of each chapter, along with complete solutions to the even-numbered problems. The total number of these carefully chosen problems is 263, and the section providing the solutions to one half of them occupies about 115 pages, that is, nearly thirty percent of the book. In general, the problem sections provide a wealth of additional results, examples, and applications, thereby enhancing the already remarkably high educational value of the main text significantly. Being largely self-contained, the book under review represents an excellent introduction to several central topics in analytic number theory. Graduate students, instructors, and even active researchers in the field can profit a great deal from the material presented in this book.

MSC:

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11A41 Primes
11B25 Arithmetic progressions
11N13 Primes in congruence classes
11N35 Sieves
11N05 Distribution of primes
11A25 Arithmetic functions; related numbers; inversion formulas
11K65 Arithmetic functions in probabilistic number theory
11N37 Asymptotic results on arithmetic functions

Citations:

Zbl 1039.11059
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