When Mathematicians Attack

LA Weekly has an interesting article about the search for a proof of Riemann's Hypothesis - the greatest unsolved math problem in existence, the mathematical Holy Grail. Even stating the hypothesis is a challenge, though in large terms it boils down to proving that prime numbers do have a purely logical distribution.

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. [apostropher: links not in original passage]

Got it? The article explores some of the work of Andrew Odlyzko, who over the past 25 years has calculated over 30 billion zeta zeros in an attempt to find one outside the critical line. As you might imagine, he is now working with inconceivably large numbers, though he believes if there are any to be found, they would probably exist in ranges well beyond current computational power, as the "wildness" in these zeros increases glacially. But as tends to happen in science, by exploring the apparent randomness to the limits of our observational abilities, a larger metapattern has begun to emerge.

Already Odlyzko’s forays into the stratospheric zone of the Riemann zeros have verified something astonishing. It turns out these zero points are not arranged randomly on the critical line. Mysteriously, they follow the same statistical pattern that physicists have found in some kinds of atomic systems — specifically, what are known as "quantum chaotic systems." Thus, what seems at first a purely abstract discovery has turned up in nature. Nobody has the slightest idea why this might be so. But the revelation suggests the incredible possibility that we might be able to find (or build) a quantum system — perhaps some bizarre kind of atom — that would prove the Riemann Hypothesis. A number of physicists are now working toward that goal.

The interplay between mathematics and the material world has fascinated philosophers and scientists alike. "God ever geometrizes," Plato declared. "All is number," Tierry of Chartres concurred in the Middle Ages. Riemann himself developed his radical non-Euclidean geometry because he was convinced there must be a geometric explanation for the force of gravity. Fifty years after his death, Einstein demonstrated the truth of that insight. The link between Riemann’s zeta zeros and quantum mechanics suggests that understanding these zeros will help to illuminate the deeper mysteries of atoms, molecules and atomic nuclei.

Though Riemann’s Hypothesis was originally stated merely as an aside, it has turned out to be one of the most profound mathematical statements ever uttered. The deeper mathematicians go into it, the more connections they continue to discover. As Sabbagh writes, "The Riemann Zeta Function extends its tentacles into so many branches of mathematics it’s impossible to say where a solution might come from."