Problems in Analytic Number Theory (Graduate Texts in Mathematics / Readings in Mathematics)

This book gives a problem-solving approach to the difficult subject of analytic number theory. It is primarily aimed at graduate students and senior undergraduates. The goal is to provide a rapid introduction to analytic methods and the ways in which they are used to study the distribution of prime numbers. The book also includes an introduction to p-adic analytic methods. It is ideal for a first course in analytic number theory. The new edition has been completely rewritten, errors have been corrected, and there is a new chapter on equidistribution.

About the first edition:

“… this monograph gives important results and techniques for specific topics, together with many exercises … it is not possible to describe adequately the wealth of material covered in this book.”

Wolfgang Schwarz, Frankfurt am Main

The chapter on sieve methods is especially good; it covers an improved Eratosthenes sieve (using the "Rankin trick") that allows you to prove Brun's estimate for the number of twin primes without using Brun's or Selberg's sieves. It then gives a very clear exposition of Brun's sieve, avoiding the combinatorial clutter that you usually see, and finishes with Selberg's sieve.

My one big gripe about the book is that there are some proofs mixed in with the problems, and unfortunately they are usually proofs of the really key results in the theory. I think this is pedagogically a weak approach, because it relegates the students to attacking less-important results. I would have been happier if the proved results had been left as exercises, although they would probably have to be split up into several steps to make them accessible to the average student.

I think problem books in mathematics are very important, and I was pleased to read this very well done book on the distribution of prime numbers.


Not all-inclusive   June 17, 2002
Ricardo R. Astudillo (Normal, IL United States)
5 out of 5 found this review helpful

Very good problems which get a mind thinking. The problems are very similar to ones that will appear on your comprehensive exams! Good practice, but there is a key element lacking in this book: motivation for solutions.

The solutions are not of great detail. Problems in this subject require advanced techniques. Unfortunately, if you have not yet had a course in the subject, these techniques appear as just "tricks" (as they are not explained with motivation). This is definitely not a book you should pick up after having only a course in undergraduate number theory. I would recommend this book only as a supplimentary to Apostol's "Intro to Analytic Number Theory", or a book of its sorts, but unless you're a complete natural, don't expect to learn analytic theory from strictly this book.

I do enjoy this book, however, and I imagine when I take the graduate course in the subject that it will be of a greater benefit, which is why I offered such a high rating.