Analytic number theory. An introductory course. (English) Zbl 1074.11001

Bateman, Paul T.; Diamond, Harold G.

Monographs in Number Theory 1. River Edge, NJ: World Scientific (ISBN 981-238-938-5/hbk; 981-256-080-7/pbk). xiii, 360 p. (2004).

This textbook steers a gentle course through multiplicative analytic number theory. Beginning with arithmetic functions and their summatory functions, it takes the student through the elementary proof of the prime number theorem (PNT), and then to Dirichlet series, the Wiener-Ikehara proof of the PNT, and then to the classical proof, with de la Vallée-Poussin’s error term. One then meets characters, Dirichlet \(L\)-functions, and the PNT for arithmetic progressions, followed by some simple oscillation theorems. The book concludes with two chapters on sieves. There are exercises scattered throughout the book, and end of chapter notes.
This book is suitable for beginning graduate students, or possibly even advanced undergraduates.

Reviewer: Roger Heath-Brown (Oxford)

MSC:

11-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11Mxx	Zeta and \(L\)-functions: analytic theory
11Nxx	Multiplicative number theory
11N37	Asymptotic results on arithmetic functions
11N36	Applications of sieve methods
11N05	Distribution of primes
11M06	\(\zeta (s)\) and \(L(s, \chi)\)

Keywords:

analytic number theory; arithmetic functions; prime number theorem; Riemann zeta-function; sieves