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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.
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