Review of Prime Obsession

By Jeffrey Nunemacher
(Department of Mathematics and Computer Science, Ohio Wesleyan University)

[The first three pages of Dr. Nunemacher's review are an account of the Riemann Hypothesis, with only passing references to the books under review (which are mine, and Dan Rockmore’s “Stalking the Riemann Hypothesis”—which, as it happens, I myself reviewed here.  Then the books and authors are briefly introduced.  Then a half page is given to Rockmore’s book.  The concluding paragraphs of the review are as follows.]

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Rockmore’s choice to include no formulas does raise the issue of how much the essence of mathematics lies in the notation used to express it.  Historically, little progress was made in mathematical areas other than geometry until modern positional notation and algebra were invented.  It is simply hard to think about and express mathematical ideas without some notation.  I don’t think one can write a clear book about the RH without being able to define the Riemann zeta function.  But why should successful books in cosmology and physics not also require some notation?  Perhaps the answer lies in the fact that a scientist employs mathematics to model phenomena, but the real subject is those phenomena themselves, which can be described generally without any mathematics.  It’s a problem with which any writer on science must grapple.

Although Derbyshire, too, begins with minimal expectations about the mathematical knowledge of his readers, he is not averse to using notation once it has been carefully explained, and his book in fact contains quite a lot of real mathematics.  Indeed, I suspect that most mathematicians can learn some new mathematics from this book (I did!)  It is definitely a book that we can recommend to our students and to nearly anyone who loves mathematics.  The only caveat is that for some of the later chapters the reader will need persistence and considerable thought.  Still, as usual in mathematics, the payoff for such attention is great, and at the end of the book the careful reader will feel a real appreciation for Riemann’s achievement.

Derbyshire writes very clearly and develops carefully in an expository fashion the tools needed to make sense of his mathematical story.  Along the way the reader learns quite a lot about the historical background of eighteenth- and nineteenth-century Europe, the world in which Euler, Gauss, and Riemann lived.  One aspect of Derbyshire’s writing that I particularly like and that should perhaps be employed more in mathematical exposition is his use of specific data and calculations to illustrate the meaning of theorems.  For example, in his demanding chapter 21, which deals with Riemann’s formula for the error terms in the approximation of pi(x) by Li(x), Derbyshire carefully computes each term in the formula and shows its contribution to pi(1,000,000).  This sort of well-explained, detailed calculation yields a real appreciation for the formula, which I suspect that many students who have seen the formula proved in a course will not have.  For some purposes calculation trumps proof.

A lot of mathematical culture is contained in Derbyshire’s book and in the extensive notes at the end of it.  It is mentioned, for example, that two different definitions for the logarithmic integral exist, an American one that is implemented in Mathematica and the European one that Riemann used.  A gallery of pictures gives human form to thirty of the main players in this story, from Euler to Odlyzko.  The book ends with Tom Apostol’s song "Where are the zeros of zeta of s?” as extended by Saunders Mac Lane and with eight pages of notes to explicate fully the lines of the song!

Both of the books reviewed are welcome additions to the popular literature of modern mathematics.  Rockmore’s book will appeal to a notation-averse reader who wishes to experience the excitement of the hunt for the solution to this important problem.  I feel greater enthusiasm for Derbyshire’s book, which also captures this excitement and which can be read profitably by any serious reader with a grasp of calculus.  I suggest that you read it yourself and recommend it to your students and colleagues.