Customer Reviews

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

5 star:		(2)
4 star:		(0)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

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.

This book concentrates on the distribution of prime numbers, omitting other analytic number theory topics such as Waring's problem and partitions. The book is split into Problems (about 170 pages) and fully worked-out Solutions (about 280 pages). The progression of topics in each chapter is well done and ramps up so the student is not overwhelmed. The book covers all the standard topics, such as Dirichlet's theorem on primes in arithmetic progressions, the Prime Number Theorem, estimates for the Riemann zeta function, functional equations, and explicit formulas.

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.