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Analytic Number Theory for Undergraduates

Heng Huat Chan
World Scientific
Publication Date: 
Number of Pages: 
Monographs in Number Theory 3
[Reviewed by
Underwood Dudley
, on

This admirable book presents, in 110 pages, an introduction to analytic number theory. The author says that he has taught the material to undergraduates over several years. The undergraduates were in Singapore, however, and probably only exceptional undergraduates in the U. S. could absorb its contents. The book is meant to be self-contained, starting with the division algorithm and greatest common divisors, but its use in a first course in number theory would be highly unusual. It contains good exercises, with no hints or solutions.

It has proofs of Bertrand’s theorem about primes between n and 2n, Dirichlet’s theorem about primes in arithmetic progression, and the prime number theorem. The exposition is clear and succinct. This is a true textbook, i.e., a book of texts that the instructor can expand on. Too many modern texts seem to try to tell the student everything, assuming that the instructor using it knows very little. This one does not, one of the reasons it is admirable.

The book is well-produced, pleasing to the eye, and has almost no typographical errors. (There is one, not at all obvious, on page 41.)

I know of nowhere else that so many important results can be found so clearly presented in so small a space. It is too bad that analytic number theory is not a standard undergraduate course, so the book could be widely used. I hope that it will be adopted, or used as a supplement, by instructors of graduate-level courses. World Scientific publishes many good books, and this is one of them.

Woody Dudley has gone through both the analytic and elementary proofs of the prime number theorem. They are both wonderful.

Facts about Integers
Arithmetical Functions
Averages of Arithmetical Functions
Elementary Results on the Distribution of Primes
The Prime Number Theorem
Dirichlet Series
Primes in Arithmetic Progression