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Publisher:

Pantheon Books

Publication Date:

2005

Number of Pages:

292

Format:

Hardcover

Price:

25.00

ISBN:

0-375-42136-X

Category:

General

[Reviewed by , on ]

S. W. Graham

01/30/2006

This is the fourth popular book on the Riemann zeta-function to appear in recent years; it joins the earlier books by Derbyshire, du Sautoy, and Sabbagh. The common theme of all of these books is the quest for the proof of the Riemann Hypothesis. This may seem like a rather technical subject for a popular work, but it ties in with questions about the distribution of the primes and the interplay between the additive and multiplicative structure of the integers. Moreover, the Riemann Hypothesis is the only problem to appear both on Hilbert's problem list of 1900 and the Clay Millennium prize list of 2000.

In style, substance, and overall quality, this book most resembles Derbyshire's book, and it is useful compare the two. Rockmore is a professor of mathematics at Dartmouth; his web page lists his research specialties as "representation theory, fast transforms, group theoretic transforms, dynamical systems, and signal processing." Derbyshire is not a professional mathematician; he is a novelist and an online columnist for *National Review*. He did, however, attempt graduate study in mathematics and got as far as a course in functional analysis. In his book, he works hard to make the mathematics of the zeta-function accessible to a reader with a high-school mathematics background. He uses a great deal of mathematical notation, although he keeps it in the even numbered chapters. By contrast, Rockmore strives to avoid introducing any equations at all, although he sometimes he will relegate them to the footnotes. These choices probably reflect the backgrounds of the two writers — Derbyshire is a frustrated pedagogue, while Rockmore has adequate alternative outlets for his teaching impulses.

A novel aspect of this book is a discussion of the correspondence between Hermite and Stieltjes on the latter's purported proof of the Riemann Hypothesis. In the 1885, Stieltjes claimed a proof of a bound for M(x) = ∑_{n ≤ x} μ(n), where μ(n) is the Möbius function. Stieltjes claimed to have proved that M(x) = O(x^{1/2}), and this implies the Riemann Hypothesis. Unfortunately, it also implies much more about the zeros of the zeta-function, and with the benefit of 120 years of hindsight, we can say that Stieltjes was almost certainly wrong. Rockmore states that there were over 400 letters exchanged between Hermite and Stieltjes. One would like to see a mathematical historian examine this closely with an eye to gaining some insight into Stieltjes' (probably misguided) approach.

Mathematicians, like everyone else, love to tell stories about each other, and the stories inevitably get exaggerated through through many retellings. One of the most often told stories in the lore of the Riemann Hypothesis is that of the meeting between Hugh Montgomery and Freeman Dyson at the Institute for Advanced Study; their revealed a connection between the zeta-function and random matrices that has turned out to be very fruitful. The story is a good one, and it has naturally morphed into many different versions. For the ultimate embellished account, see the "screenplay version" of Hayes.

Derbyshire and Rockmore both give accounts of this story, but their accounts differ in several respects. Derbyshire places the story in 1972; Rockmore in 1974. In Rockmore's version of the conversation,

Montgomery began to explain his recent results on pair correlation, and Dyson stopped him short — "Did you get this?" he asked, writing down a particular mathematical formula. Montgomery almost fell over in surprise...

According to Derbyshire, Montgomery says

I told him I was working on the differences between the non-trivial zeros of Riemann's zeta function, and that I had developed a conjecture that the distribution function for those differences had integrand 1 - (sin(πu)/πu)^{2}. He got very excited. He said: "That's the form factor for the pair correlation of eigenvalues of random Hermitian matrices!"

On balance, Derbyshire's version is more reliable than Rockmore's. The 1972 date can be confirmed from checking the volume where Montgomery's article appears. The suggestion that Dyson would guess the form factor with minimal prompting stretches credibility. Neither author identifies the source of their stories, but Derbyshire has informed me that his version is a transcript of remarks that Montgomery made in a lecture at a workshop in Seattle in August 1996.

There is another aspect of the story of pair correlation that Rockmore and all the other popular authors on the Riemann-zeta function have missed. That is the fact that it took about 10 years for people to really importance of Montgomery's discovery. Goldston observes that the first few papers on pair correlation attracted little attention. "Then, in the early 1980's, everything changed: Odlyzko computed statistics on the zeros and convinced even the most skeptical that after almost a century of intensive study a new, unsuspected, and fundamental property of the zeta-function had been discovered."

The connections between the zeros of the zeta-function and random matrix theory have become the most active and exciting threads of research in the hunt for the Riemann hypothesis. Rockmore devotes four chapters at the end of his book to various aspects of this research. He discusses the work of Sarnak and Katz on analogous results for function fields. He also discusses work of Tracy, Widom, and Deift that connects the distribution of eigenvalues of random matrices to properties of permutations. This chapter has the engaging title "God May Not Play Dice, but What About Cards?"

While books such as this one are aimed at mathematical amateurs, perhaps the most appropriate audiences are undergraduate mathematics majors and professional colleagues in fields other than mathematics. For those audiences, I would recommend Derbyshire's book as a first choice. However, if one's budget (in terms on time and money) permit, I would also recommend this book, particularly in the last four chapters.

**References:**

John Derbyshire, *Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics*, Joseph Henry Press, 2003. See the review by S. W. Graham.

P. Deift, *Integrable systems and combinatorial theory*, Notices of the American Mathematical Society **47** (2000) 631-640.

D. Goldston, *Notes on pair correlation of zeros and prime numbers*, arXiv:math.NT/0412313.

B. Hayes, *The spectrum of Riemannium*, American Scientist **91** (2003) 296-300.

H. L. Montgomery, *The pair correlation of zeros of the zeta function*, Proceedings of the Symposium of Pure Mathematics: Analytic Number Theory **24**. AMS, Providence, R.I., 1973, 181-193.

A. M. Odlyzko, *On the distribution of spacings between zeros of the zeta function,* Math. Comp. **48** (1987) 273-308.

Karl Sabbagh, *The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics*. Farrar, Straus, and Giroux, 2003. See the review by S. W. Graham.

Marcus Du Sautoy, *The Music of the Primes*, Harper Collins, 2003. See the review by Stacy Langton.

S.W. Graham is currently a Professor of Mathematics at Central Michigan University. He can be reached at sidney.w.graham@cmich.edu.

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