Three Books on Riemann Hypothesis

My first review will be of three semi-popular books about the Riemann hypothesis: Prime Obsession by John Derbyshire, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics by Marcus Du Sautoy, and The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics by Karl Sabbagh.

Recently there has been a spate of semi-popular books about the Riemann Hypothesis. This is doubtless due in part to the fact that several of the most famous problems of modern mathematics such as the Four-Color Map Theorem, Fermat's Last Theorem, and the Poincare Conjecture have now been solved, leaving the Riemann Hypothesis as the most famous problem standing. However, writing a semi-popular book about the Riemann Hypothesis is an intimidating mission. Unlike the Four-Color Map Theorem and Fermat's Last Theorem, it is difficult to explain to an educated layperson what the theorem states, or why it is important. Even the statement of Poincare Conjecture is easier to comprehend.

John Derbyshire's Book, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, is the best of the lot. He sets himself the daunting task of explaining virtually all of the major mathematical ideas needed to understand the statement of the RH, its relation to the distribution of prime numbers, and some of the major methods that have been used to attack the problem, in a book designed for an otherwise educated person who is ignorant of mathematics from high school algebra on. It sounds to me that this goal must have been imposed the publishers, because whatever the talents of the expositor, it is prima facie impossible to bring anyone but a latent mathematical genius on such a trip in the confines of a single 422-page book. However, what Derbyshire does, and does brilliantly, is to explain the RH to someone who has understood two years of college calculus, or the equivalent. The reader who has experience with integrals and infinite series should be able to follow the exposition.

The Reimann Hypothesis and its relation to the distribution of primes belongs to the branch of mathematics called analytic number theory. This subject is not easy to write about. I took a reading course in analytic number theory in graduate school. I was intrigued by the subject, but became discouraged when I found the text, by a famous researcher in the field (who shall remain nameless) riddled with errors. I ended up going into another specialty. Now that I have read Derbyshire's book, I'm tempted to read more. In addition to the mathematical exposition, Derbyshire quickly and deftly sketches the political and social milieu and the personalities involved in the development of the RH and the search for its solution.

I bring up one quibble because it relates to the first chapter, and might cause a reader to give up. Derbyshire introduces the harmonic series (and its divergence) by asking the reader to imagine constructing a bridge out of playing cards. It turns out that the n-th card from the top of this bridge can extend 1/(n – 1) card length from the card above it, so that the span of the entire n-card bridge is 1 + 1/2 + … + 1/n. I've seen this before, and it is cute, but it is not easy. There are easier ways to introduce the harmonic series.

Marcus du Sautoy's book, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, covers much of the same territory as Derbyshire's book, but goes into somewhat less mathematical detail. Du Sautoy is a professor of mathematics at Oxford, and an excellent writer. I recommend this book for the poetry of the language and the vividness of the stories of the mathematicians involved in the story. It is wonderful to read a book by a first-rate mathematician who is also a first-rate storyteller.

In Karl Sabbagh's book, The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics, the author, like John Derbyshire, attempts to explain the RH to the mathematically unsophisticated reader. In this case, the mathematical basics are covered in a series of appendices, called "Toolkits". This book was something of a disappointment. First, the author comes across as a journalist rather than as an advanced amateur mathematician (like Derbyshire) or a professional mathematician (like du Sautoy). The writing has more of a superficial feel to it, where more tends to be made of the physical appearance or personal idiosyncrasies of mathematicians rather than their ideas. In addition, Sabbagh spends much of the book conversing with and about Louis de Branges, who has claimed to have a proof of the Riemann Hypothesis. It is true that de Branges is a respected mathematician who solved an important long-standing problem, the Bieberbach Conjecture. However, very few mathematicians credit his claims to have, or be close to, a proof of the Riemann Hypothesis. Sabbagh is obviously charmed by de Branges, and spends, in my opinion, far too much time on this player who seems to deserve at most a short footnote in the story.