Prime suspects
What is the most down-to-earth yet significant problem in mathematics? In the beginning, you learn to count -- 1, 2, 3, 4, 5 -- and to add these numbers. Next comes multiplication, a way to batch-process multiple additions. A great observation can then be made: Some of the counting numbers can be obtained by multiplying two smaller counting numbers together -- like 6, which is 2 times 3 -- but others -- like 5 -- cannot. Irreducibility is key in mathematics and science. The numbers that are not reducible to smaller factors are called prime numbers.
How many prime numbers are there? Are they relative anomalies or common? Euclid supplied a proof that the number of primes is infinite: There is no largest prime. All right, if we are given a number, however high, can we find a way to tell how many primes there are below that number without actually counting them?
The answer, at least in a first approximation, turns out to be surprisingly simple. Dividing the number you are interested in by its natural logarithm (there is a logarithm button on many calculators) gives the number of primes that will be found beneath it. This result is one of the jewels of mathematics, the “prime number theorem.”
Jacques Hadamard and Charles-Jean de la Vallee Poussin proved the prime number theorem in 1896, based on an approach suggested by a nearly telegraphic, eight-page paper by Bernhard Riemann (1826-1866) published in 1859. This approach uses what has come to be known as the Riemann hypothesis, which at first glance might seem to be merely technical and not something that would prompt, more than 140 years later, the appearance of four popular books.
The paper introduced “Riemann’s zeta function,” which takes on various values. Riemann said that if one graphs the points where the zeta function equals zero, all the points will lie on a specific line. What Hadamard and De la Vallee Poussin proved was that they all lie inside a fairly wide band on either side of the line. Proof of the full Riemann hypothesis would give us a much more precise prime number theorem and a lot of information about prime numbers, even when they are unreachably large.
Number theorist Marcus du Sautoy’s book “The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics” compares the pattern the primes make to music. At first they seem to appear at random, like a kind of noise, but under analysis display a more musical structure.
Du Sautoy writes: “Mathematics is a creative art under constraints -- like writing poetry or playing the blues. Mathematicians are bound by the logical steps they must take in crafting their proofs. Yet within such constraints there is still a lot of freedom. Indeed, the beauty of creating under constraints is that you get pushed in new directions and find things you might never have expected to discover unaided. The primes are like notes in a scale.”
Du Sautoy shows how computers are used to discover reams of detail about the primes and how this detail is important to Web commerce. His account of current work takes us as close to the frontier as we can get without a passport.
John Derbyshire’s “Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics” is a more difficult book but is even more rewarding for those up to the challenge. Energetic and conversational, it puts one at ease. In even-numbered chapters he gives a historical overview and biographical sketches of Riemann and those who followed him, while in odd-numbered chapters his mathematical exposition is clear. Derbyshire occasionally sideswipes calculus but usually succeeds in avoiding it.
Derbyshire, who studied mathematics and linguistics at London University, has worked as an investment banker and computer programmer. In 1996 he published a comic novel, “Seeing Calvin Coolidge in a Dream.” His day job is writing commentary for National Review, the New Criterion and the Washington Times. Derbyshire’s attempt to take nonmathematicians into this subject had me on the edge of my seat. Was he really going to introduce Moebius inversions in polite company? He did, and I found his treatment, and his chutzpah, consistently interesting. His account of what has happened in the last 30 years is sure-footed and perceptive.
Late in the book, Derbyshire wonders whether a proof of the Riemann hypothesis is near, saying, “I am, therefore, going to stick my neck out and say that I believe a proof of the RH to be a long way beyond our present grasp. Surveying the modern history of attempts on the RH is something like reading an account of a long and difficult war. There are sudden surprising advances, tremendous battles, and heart-breaking reverses. There are lulls -- times of exhaustion, when each side, ‘fought out,’ does little but conduct small-unit probes of the enemy defenses. There are breakthroughs followed by outbursts of enthusiasms; and there are stalemates followed by spells of apathy.”
Karl Sabbagh also takes a shot at explaining this fundamental problem in “The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics.” Of the world of mathematicians, Sabbagh writes, “For me, it will be as if I am describing a remote tribe whose customs and language are unfamiliar to the reader, but whom I understand enough to convey something of their inner and outer lives.” It is not unheard of for anthropologists to spend time with remote tribes without fully understanding what they are seeing. Much of Sabbagh’s book assembles quotations from people he interviewed, the sort of talk that you might hear at a mathematics tea or around the edges of a conference. A journalistic effort along these lines could capture the chaos, guesses and gossip that accompany the search for a Riemann solution, but Sabbagh doesn’t bring enough mathematical background to the enterprise.
He features a “proof” offered by Louis de Branges of the Riemann hypothesis, displaying it in an appendix, and on this point I fear he was hornswoggled by one of the locals. A review article on the Riemann hypothesis in the March Notices of the American Mathematical Society mentions just about everything but De Branges’ proof. De Branges is an interesting character -- maybe even a dark horse to prove the Riemann hypothesis. But Sabbagh betrayed his subject when he made book on De Branges, perhaps thinking it would make a great story if he were right.
Keith Devlin’s excellent book, “The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time,” puts proving the Riemann hypothesis first. Devlin, who appears on NPR’s Weekend Edition as “the Math Guy” and is the author of numerous books, is a pro. He is savvy about knowing what he might have a chance of explaining and what is likely to get him into trouble. The examples he uses to explain key ideas are often exceptionally well-chosen, and if you want a concise introduction to the Riemann hypothesis, this is your book.
Mathematician Andrew Odlyzko, quoted by Derbyshire, says: “It was said that whoever proved the Prime Number Theorem would attain immortality. Sure enough, both Hadamard and de la Vallee Poussin lived into their late nineties.” But proposing a corollary involving the Riemann hypothesis, he adds, “should anyone manage to actually prove its falsehood -- to find a zero off the critical line -- he will be struck dead on the spot, and his result will never become known.” This is gallows humor, because Odlyzko uses computers to look for zeros that might contradict the Riemann hypothesis. Perhaps he thinks the risk is worth it-- that’s how important the Riemann hypothesis is to mathematicians.
