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The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics
The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics
Karl Sabbagh
Farrar, Straus and Giroux, 2004
342 pp., $16.00

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Karl-Dieter Crisman


Three Prime Numbers Go into a Bar

The curious history of the Riemann Hypothesis.

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Editor's note: Science in Focus is on vacation in August, resuming our regular schedule in September. Meanwhile, we're going to the archives for science-related pieces from the pages of Books & Culture. This week we're featuring a piece by Karl-Dieter Crisman from the January/February 2009 issue.

Ask the next person you meet what the greatest unsolved problem in mathematics is, and you'll be lucky if the worst you get in return is a quizzical stare. A few might mention the search for a grand unification of modern physics, but that's not really right, since even the most abstract mathematical physics is (in theory, at least) connected to the real world.

But most people probably will say, "I don't care," if they respond at all. So it may be more than a little surprising to discover that in roughly the last five years no fewer than seven books, all (ostensibly) aimed squarely at the layman, treat precisely this question.

One, by veteran math popularizer Keith Devlin, attempts to describe all seven of the Millennium Prize problems [1], for the solution of which the Clay Institute for Mathematics (www.claymath.org) has recently offered million-dollar prizes. Two more describe the intense human drama of the recent solution of one of these, a century-old problem known as the Poincaré Conjecture. [2]

Given the compelling math, and the story's culmination in the explicit rejection of the most prestigious prize in mathematics by Grigory Perelman, the Russian genius behind the solution, [3] the choice of this topic for a book makes sense. Yet it is telling of the consensus in mathematics as to the answer to our little question that all four of the other books tackle a different Millennium Prize problem, known as the Riemann Hypothesis (or RH, for short). We here review the two of these of greatest interest to Books & Culture readers.

With the history of post-Napoleonic Europe as a lush backdrop, John Derbyshire's Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics attempts to unpack the problem itself; on the other hand, Karl Sabbagh's The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics has a far lighter treatment of the math, focusing on "the humanity of mathematicians," in particular those mathematicians working on the RH on the cusp of the 21st century. Both achieve their major goals, though each book has certain deficiencies I'll outline later. [4]

Even at the risk of pedantry, it seems strange to go on without briefly describing what the Riemann Hypothesis is. Here is the Clay Institute's own description of the RH:

Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern, however the German mathematician G.F.B. Riemann (1826–1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function "?(s)" called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation ? (s) = 0 lie on a straight line. This has been checked for the first 1,500,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of primes.

Even most professional mathematicians could only add the definition of the Riemann Zeta function and what the word "interesting" means; the "close relation" is the heart of this most difficult question.

Given the spate of popular math books of the last two decades, and the relative inaccessibility of the problem, why would anyone write or read about it? In Sabbagh's book, the interest is to some extent in TV-style voyeurism. Lest anyone think I mean a reality-show race to wealth, keep in mind that the problem is 145 years old, has been seriously worked on for over a century, and will almost certainly not be solved anytime soon. [5] Instead of Survivor, the researchers in The Riemann Hypothesis are on All My Children. We hear shouts of controversy over two claims on the same proof of an important theorem by Erdös and Selberg; we witness a chance conversation at Princeton that yields one of the main means of attacking the RH; we see the late entrance of the "lovely" Fields medalist Alain Connes as a knight to (possibly) save the day. There is even a black sheep in the family, Louis de Branges; his story begins like the rest, but becomes a tragic portrait of a man "for whom there is nothing in the world but mathematics and persecution," for even if his latest proof is correct, he has cried wolf too often for anyone to listen any longer.

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