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The Prime Number Theorem

G. J. O. Jameson
Publisher: 
Cambridge University Press
Publication Date: 
2004
Number of Pages: 
x + 252
Format: 
Paperback
Series: 
London Mathematical Society Student Texts 53
Price: 
29.99
ISBN: 
0-521-89110-8
Category: 
Textbook
[Reviewed by
David Bressoud
, on
10/22/2005
]

The prime number theorem has always been problematic for me. I love to teach it for this is one of the great mathematical accomplishments of the 19th century, but the proof does not fit comfortably into either the undergraduate or graduate curriculum. There are too many specialized results and the proof is too involved to do an adequate job within the context of a course in complex analysis, undergraduate or graduate. To teach it the way I prefer, following Davenport and Montgomery’s treatment in Multiplicative Number Theory [1], is out of the question for an undergraduate seminar. There is simply too much ground to cover. I have used the context of an independent study course to help an undergraduate work through D. J. Newman’s simple proof [2,3], but that proof is so magical that I question whether anything was really learned.

The prime number theorem is too specialized, too particular to fit comfortably within a graduate course in number theory. A single semester necessarily emphasizes algebraic number theory. A second graduate semester on number theory, coming after the completion of a full year of graduate complex analysis, can give full attention to the prime number theorem, but now suddenly there is not enough. The proof of the prime number theorem is just an introduction to analytic number theory. It has lost its rightful grandeur.

Jameson’s book seeks to correct this. It is in a series designed for advanced undergraduates and beginning graduate students (admittedly, undergraduates in the British university system who are usually far advanced of those in the United States).  It presents a complete, self-contained course that could be taught to students who have not studied complex analysis, though intimate familiarity with the complex plane is assumed.

This book presents all three of the main proofs: the traditional Hadamard-de la Vallée Poussin proof (as refined since then), the Erdös-Selberg elementary proof (elementary but far from simple), and Newman’s magical proof that first appeared in the Monthly in 1980. Furthermore, Jameson does Newman’s proof the right way. That is to say, he embeds it into the development of the traditional proof, enabling a direct comparison and showing exactly what steps Newman managed to shortcut.

Two aspects of this book particularly endear it to me. One is the discussion of applications of the prime number theorem. There is the obvious but important corollary that for any c > 1 and x sufficiently large (dependent on c), there will always be a prime in the interval (x , cx). There is also a derivation of the formula for the number of integers less than x with exactly k prime factors (with or without repetition counted). This result is important in analyzing factoring algorithms whose average speed depends on the number and size of the prime divisors one is likely to encounter. The other aspect is the extended discussion of error estimates and the Riemann hypothesis. As a student of Emil Grosswald, I spent many hours entranced by his explanations of what was known about the behavior of zeta in the critical strip. There are echoes of those lectures here.

This book is not perfect. Because so little is assumed and so much groundwork must be laid, the early chapters are devoted to proving many results whose purpose is unclear. Jameson is not able to carry the structure of Davenport and Montgomery in which each chapter was based on a concept. I particularly miss the Davenport and Montgomery discussion of Riemann’s original memoir which provided the outline for how to prove the prime number theorem, an outline that Davenport and Montgomery follow in their exposition. Jameson includes only a fleeting reference to this memoir. I was also disappointed that the functional equation for the zeta-function is stated but, since it is never actually needed, not proven or discussed.

 Jameson chooses to put the proof of Dirichlet's theorem on primes within an arithmetic progression after completing the proof of the prime number theorem. This has the pedagogical advantage that it enables a seamless transition from the proof that there are infinitely many to the derivation of the asymptotic formula. It is an intriguing choice, and I am curious to see how it works.

I am looking forward to trying this book in a senior seminar. It will take a group of very exceptional undergraduates, but I think I can round them up, especially when I can promise them that they will learn the proof of the prime number theorem.

References:

[1] Davenport, Harold, Multiplicative Number Theory, second edition, revised by Hugh L. Montgomery, Springer-Verlag, New York, 1980.

[2] Newman, D. J., Simple Analytic Proof of the Prime Number Theorem, American Mathematical Monthly, 87 (1980), 693-696.

[3] Zagier, D., Newman’s Short Proof of the Prime Number Theorem, American Mathematical Monthly, 104 (1997), 705-708.

Note: This book is also available in hardcover, ISBN 0-521-81411-1, $70.00 (not seen).


David M. Bressoud is DeWitt Wallace Professor Mathematics at Macalester College in St. Paul, Minnesota.

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