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An Introduction to the Analytic Theory of Numbers

Raymond Ayoub
Publisher: 
American Mathematical Society
Publication Date: 
1963
Number of Pages: 
379
Format: 
Electronic Book
Series: 
Mathematical Surveys and Monographs 10
Price: 
93.00
ISBN: 
9781470412388
Category: 
Monograph
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on
09/13/2017
]

This is a comprehensive introduction to analytic number theory published in 1963. It is dated in the sense that we have better (or at least slicker) proofs for most of its results, but the proofs given here are still well worth studying. The book appears to have been aimed at graduate students; the prerequisites are small, but the proofs are intricate and require a lot of skill to work through. The exposition is masterful, there are lots of interesting side paths, and the chapter end-notes are interesting and helpful. There are extensive sets of exercises in each chapter. These are generally easier than the material in the body; many of them show what can be done with simpler methods, some follow up sidelines mentioned in the body, and a few extend intermediate results.

One of the charms of this book for the serious student is the variety of approaches and proofs. The book starts out with a proof of Dirichlet’s theorem on primes in arithmetic progressions (using a complex-variables proof that the \(L\)-series does not vanish at 1), then has four different proofs of the Prime Number Theorem. There’s a lengthy chapter on asymptotics of the partition function, including the Hardy–Ramanujan asymptotic estimate and asymptotic series, an excursion into the Dedekind \(\eta\)-function, and Rademacher’s convergent series. There’s a lengthy chapter on the solution to Waring’s problem (this is part of the book that is most out of date, where the greatest simplifications have been made). Finally there is a classical exposition of the class number of quadratic fields in terms of the Dirichlet \(L\)-series.

The book is aimed at proving particular results, while developing methods that can be used more generally. It makes good use throughout of Tauberian theorems and of the Hardy-Littlewood circle method. The most conspicuous omission is sieve methods; these did not amount to much when the book was written, but have had a tremendous growth since then.

Despite its age, this book still stands out for its breadth of coverage. I don’t know any other introductory book that covers as wide a variety of methods and results as this one. True, we have slicker and more direct proofs for many of the results here (and many of these are in the book’s nearest competitor, Newman’s quirky Analytic Number Theory). To many mathematicians (and many textbook writers) analytic number theory means the distribution of prime numbers, and several introductory texts cover just that. Iwaniec & Kowalski’s Analytic Number Theory (AMS, 2004) does have very broad coverage, although it is organized by method and tool rather than by result, and in my opinion is not an introduction. I think the best overall introductory book is Apostol’s An Introduction to Analytic Number Theory; it has excellent explanations, and covers a variety of topics, although it does not have the breadth of the present book. It is also unusual in being aimed at undergraduates. A well-regarded and thoroughly-modern introductory book, that I have not seen, is Overholt’s A Course in Analytic Number Theory.

 

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Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.

I. Dirichlet's theorem on primes in an arithmetic progression

II. Distribution of primes

III. The theory of partitions

IV. Waring's problem

V. Dirichlet L-functions and class number of quadratic fields