Prime Obsession

It’s been called the hardest problem in mathematics, and also the most important. Three books have recently been published about it and dozens of the world’s most brilliant mathematicians are devoting themselves to it. A million- dollar prize awaits the person who solves it. Known as the Riemann Hypothesis, no mathematical problem inspires such fear and awe — it is said that some mathematicians would sell their souls for the answer.

A decade ago, mathematics splashed onto front pages the world over when Andrew Wiles announced that he had solved another famous problem known as Fermat’s Last Theorem. The public interest in Wiles’ solution stunned the mathematics community — a member of their notoriously nerdy fraternity was now being asked to pose for Gap ads. Suddenly, “It felt almost sexy to be a mathematician,” writes Oxford don Marcus du Sautoy in his new book, The Music of the Primes (HarperCollins). And next month’s publication of David Foster Wallace’s Everything and More, a 300-page rumination on transfinite numbers, seems set to propel math into the lexicon, and onto coffee tables, of the literary cool set.

With a pedigree linking many of the greatest names in the field, the Riemann Hypothesis runs like a river through vast swaths of seemingly distinct mathematical territory. Andrew Wiles himself has compared a proof of this proposition to what it meant for the 18th century when a solution to the longitude problem was found. With longitude licked, explorers could navigate freely around the physical world; so too, if Riemann is resolved, mathematicians will be able to navigate more fluidly across their domain. Its import extends into areas as diverse as number theory, geometry, logic, probability theory and even quantum physics.

The Riemann Hypothesis is a proposal about prime numbers, the atomic elements of the number system. Primeness is one of the most essential concepts in mathematics, for primes — 2, 3, 5, 7, 11 and so on — are numbers that cannot be broken into any smaller elements. All other integers can be built up by multiplication of these basic units. So, for example, 6 is built up from 2 x 3, 15 from 3 x 5, 49 from 7 x 7. In his book The Riemann Hypothesis (FSG), science writer Karl Sabbagh makes an analogy between numbers and molecules. All of the vast plethora of molecules that inhabit our world, everything from salt and ammonia to hemoglobin, are made up of the basic elements of the periodic table — carbon, hydrogen, oxygen and so on. As Sabbagh notes, the primes may be seen as the periodic table of the number system. Yet where the elements follow a clear pattern, the primes seem to be distributed randomly.

To mathematicians, randomness is anathema. As du Sautoy writes, they “can’t bear to admit that there might not be an explanation for the way nature has picked the primes.” That would be like “listening to white noise”; what mathematicians crave above all else is harmony. They want, they need, they demand a pattern behind the apparent chaos. Du Sautoy quotes the great French mathematician and physicist Henri Poincare: “The scientist does not study nature because it is useful, he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing and if nature were not worth knowing, life would not be worth living.”

For many mathematicians life would not be worth living if the organization of the primes did not ultimately conform to some beautiful underlying order. The Riemann Hypothesis proposes what that order might be. One of the most innovative mathematical thinkers of all time, Bernhard Riemann was a sickly German genius of the mid–19th century who also developed the non-Euclidean geometry on which Einstein based his general theory of relativity. Riemann’s life and work form the subject of John Derbyshire’s touching biography Prime Obsession (Joseph Henry Press). Though he died at 39 and his collected works amount to a single slim volume, virtually every paper Riemann wrote revolutionized a different branch of mathematics.

Riemann’s Hypothesis is not easy to state in any language. In essence it links the distribution of prime numbers to a complicated equation called the Riemann Zeta Function. For some values this equation equals zero, and it turns out there are an infinite number of such values, which mathematicians refer to as the “zeros” of the zeta function. Riemann demonstrated that there is a beautiful and unexpected link between these “zeros” and the pattern of the prime numbers.

Each Riemann zero can be represented as a point on something called the complex plane, one of mathematics’ most truly enchanted places. Formed from the intersection of the “real” and the “imaginary” numbers, the complex plane is also where the fabled Mandelbrot Set lives. To his astonishment, Riemann discovered that on this plane the zeta-function zeros seemed to lie in a strict vertical line, which is now called the critical line. Why this might be so is one of the deepest questions in mathematics. It was Riemann’s intuition, his hypothesis, that all the zeta zeros must lie on this line.