Colloquium Publications. American Mathematical Society 53. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3633-1/hbk). xi, 615 p. $ 79.00 (2004).
The first two sentences of the preface describe the goals of the authors: “This book shows the scope of analytic number theory in classical and modern directions. There are no division lines; in fact our intent is to demonstrate, particularly for newcomers, the fascinating countless interrelations.” Compared with some more or less recent monographs on various branches of analytic number theory such as the zeta- and $L$-functions with applications to the distribution of primes, exponential and character sums, sieve methods, automorphic functions with spectral theory, Diophantine problems, the circle method, and so on, the range of topics in this comprehensive volume is remarkably wide. Indeed, all the topics mentioned above are covered, among others. There are 26 chapters starting from “Arithmetic Functions” and ending with “Central Values of $L$-functions”.
Instead of giving the long list of all these 26 headings, let us try to classify these under fewer headings. The reviewer ended up with the following classification (the number after each topic gives the number of those chapters falling under that heading; this decision was not always easy): elementary methods and sieves (4), character and exponential sums (4), arithmetic functions (7), zeta- and $L$-functions together with Dirichlet polynomials (6), automorphic functions and spectral theory (3), analytic algebraic number theory (2). The emphasis in this monograph is on ideas and methods rather than on “records” in various problems, and the proofs that are chosen to be presented are usually reasonably short, and also illuminating in some general sense.
As to the audience of this text, the authors say having had especially graduate students in mind. True, though the presentation in principle starts more or less from scratch, some sophistications can be best appreciated and enjoyed against the background of certain preliminary familiarity. Here and there one may find novel or unusual lines of argument; for instance, the evaluation of quadratic Gauss sums in Theorem 3.4 is based on van der Corput’s lemma on exponential sums rather than on Poisson’s summation formula. Naturally, it is impossible to keep a treatise of this size and complexity completely free from errors and misprints, but these are mostly easy to spot and correct.
The authors are active researchers with a lot of experience and deep insight, and their creative attitude makes reading particularly rewarding. The very choice of material reflects a modern “global” view of analytic number theory, and hence, even with its encyclopedic wealth of topics, the flavor of this book is not at all encyclopedic. It can be warmly recommended to a wide readership, at various levels of expertise, interested in analytic number theory with its many fascinating “eternal” (?) problems and numerous links to other branches of mathematics.