Book review - The Mathematical Unknown

by John Derbyshire
Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
Joseph Henry Press, 412 pages
reviewer James Franklin
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With the proof of Fermat's Last Theorem in 1994, Derbyshire says, "the Riemann Hypothesis is now the great white whale of mathematical research". Even before that, it was regarded by mathematicians as the more significant problem - though not as old as FLT, it is more central to mathematics and probably a good deal harder.

And harder to explain. Of the two new books offering an account for a popular audience, Prime Obsession and Karl Sabbagh's The Riemann Hypothesis (Farrar, Straus and Giroux, $25), Derbyshire's offers the better insight. Sabbagh's is a well-written book with interesting stories from mathematicians working in the field, but Derbyshire is a talented expositor determined to make the reader understand some serious mathematics. A general reader with some memory of high school algebra who is willing to concentrate will come away with a grasp of what the problem is and why insiders are excited. Mathematicians in other fields will deepen any superficial understanding they may have, as well as picking up some new ideas on how to explain mathematical ideas.

The importance of the Riemann Hypothesis comes from its close connection to one of the most basic phenomena connected with numbers, the distribution of primes. At first glance, numbers all look much the same except for their size. They are not. Twelve eggs can be arranged as a rectangle of 6 eggs by 2, or 3 by 4. That cannot be done for 11 or 13 eggs. 11 and 13 are primes, numbers that cannot be written in any way as a product of smaller numbers. The primes less than 50 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

Between 980 and 1000, the only primes are 983, 991 and 997. Between 9980 and 10000, there are none. It can be seen that the primes thin out as we go further along the numbers (though they never run out). There is, however, an irregularity or jaggedness to the way they thin out. It is believed, though not proved, that there are an infinite numbers of prime pairs - that is, however far out we go, there is always an occasional pair of odd numbers only two apart that are both primes (like 41 and 43). On the other hand, there are indefinitely long stretches of numbers with no primes at all. It is easy to understand why: to get a stretch of, say, 100 numbers without a prime, multiply together all the numbers from 1 to 101 and call the answer x. Then the 100 numbers x + 2, x + 3, x + 4, ... , x + 101 are all non-primes (since x + 2 is divisible by 2 because x and 2 both are, x + 3 is divisible by 3, and so on). Thus the sequence of prime numbers, though it is a matter of absolute necessity and the same in all possible worlds, has the interplay of overall orderliness with local irregularity that we are accustomed to in sequences of throws of dice and coins. Einstein may or may not have been right to say that God does not play dice with the universe, but He certainly does not play dice with the primes. Numbers are not subject to chance or to any will, human or divine. That is what gives questions about the distribution of primes their peculiar fascination.
The local irregularities make it hard to answer the "big-picture" question: at what rate do the primes thin out? What is the average density of primes - if we take a block of 1000 consecutive numbers around, say, 1 trillion (1,000,000,000,000), what proportion of them should we expect to be primes? The answer is given by the celebrated Prime Number Theorem, whose proof was one of the glories of late nineteenth century mathematics. The density of primes near the number N is about 0.434/log(N) (where log(N), called the logarithm of N, is the number of digits in N). The important item in this formula is not the 0.434, which relates to the fact that we have decided to write numbers in base 10 (if we wrote them in base 2 as computers do, the number of digits in N would be greater). The significant thing is that the density of primes thins out logarithmically - since 1 trillion has twice as many digits as 1 million, the density of primes around 1 trillion is half what it is around 1 million.

Riemann's 1859 paper, in which he introduced his Hypothesis, is a bold series of moves which gives a formula not only for the average density of primes but for all the irregularities as well. Late in his book, Derbyshire ambitiously but successfully unpacks this short and difficult paper, and explains how Riemann gives an exact formula for the deviations from the average density, so that one can calculate exactly how many primes there are in any block of numbers. The formula has a downside, however. It expresses the answer in terms of some mysterious entities called the zeros of the Riemann zeta function. There are an infinite number of these beings, and the Riemann Hypothesis says of them, "All the non-trivial zeros of the Riemann zeta function have real part one half." Explaining from a standing start what the Riemann zeta function and its zeros are in only half a book is not easy, and Derbyshire proves himself a leading mathematical communicator in being able to do it. "If you don't understand the Hypothesis after reading my book", he says, "you can be pretty sure you will never understand it." He is right.

The book is not all tough mathematics. Included, for example, is the bizarre connection between the way the zeros of the zeta function occur and the way some quantum mechanical systems are spaced. There is something on the use of prime numbers in internet security. There is some history, including the little there is to know about Riemann himself. He was pious, shy, depressed, and died of tuberculosis aged not quite forty. On the real world, his impact was minimal. When he went through the door into his study and tapped into the abstract world, he made enormous advances in several different mathematical fields. "Riemann's mathematics has the fearless sweep and energy of one of Napoleon's campaigns."
Derbyshire handles with kid gloves, as well he might, a question unavoidable when talking about an unproved hypothesis, that of probabilistic reasoning in pure mathematics. He writes that "Everybody knows that in mathematics you must prove every result by strict logic." That is true in the sense that a strict proof of everything is sought, but it is not true if it means that anything not proved is not yet part of mathematics. If that were true, there would be no book about the Riemann Hypothesis, since it is not proved. So is the evidence for its truth good? Should we gather evidence for and against it, as if it were a defendant in a court of law? Since the Hypothesis has the same logical form as "All swans are white", the most direct sort of evidence comes from calculating the zeros and checking if their real part is indeed a half. The zeros are ordered, so one can speak of the first one, second one, and so on. It was shown in 1903 that the first 15 zeros do have real part a half, and both people and machines have been busy since. 50 billion is announced, but it is hard to keep track, since there is a co-operative project using spare computer time that claims to be knocking over a billion zeros a day. It is one of the largest inductions in history. In such abstract areas, however, it is not surprising that there are more subtle reasons bearing on the question. Most experts are firmly convinced that the Hypothesis is true, but there still are a few sceptics. There is just a little reason to think that though there are no small counterexamples (zeros with real part not a half), there could be some very large ones, ones far beyond the reach of any feasible calculation. There is something to be said for the opinion of the mathematician George Polya, that pure mathematics is the best place to appreciate probabilistic reasoning. For in mathematics, there are no distractions from subjective factors or laws of nature. The hypothesis confronts the evidence in pure logical space.

Will the Riemann Hypothesis be proved soon? Derbyshire takes the risk of making a fool of himself and puts his prediction on the line: no. The ideal reader of his book, then, is an obsessive fifteen-year-old genius like the young Gauss, who often spent an "idle quarter of an hour" tallying the primes in blocks of a thousand. The problem will most likely still be there when that reader is old enough to tackle it.


Updated: 14 Dec 2007