A decade later, he and a colleague invented a much better algorithm, and in one mighty bound Odlyzko leapt up the critical line to the 1020 Riemann zero. Now he was operating in a zone where no human had ever set foot. “To calculate all the zeros up to this point would require more computational time than there’s been in the history of humanity,” Odlyzko notes. Another jump brought him to the 1021 zero, then the 1022, and finally the 1023. At each step, billions more zeros fell to his computational scythe. But still the wild things eluded him.
“If I could do 1050, I would do it,” Odlyzko says. Yet he now suspects that even this would not be good enough. So slowly does the “wildness” grow, he believes that if exceptions exist off the critical line, they will probably not be found until around the 10100 Riemann zero. At present that region is beyond any currently conceivable computing power. And even if he could reach that high, Odlyzko notes philosophically that “counterexamples are likely to be extremely rare.” His chances of finding one are essentially nil. Nonetheless, he keeps on climbing.
In the mathematical universe, Odlyzko is applying what is known as a brute-force approach — the more computer power he can bring to bear, the more zeta zeros he can calculate, hence the more likely he is to find aberrations. On the whole, mathematicians disapprove of computational approaches; a widespread attitude, especially among older mathematicians, holds that the only “real” proof is an analysis derived from fundamental axioms. All else is hack work.
Odlyzko is aware of this bias among his colleagues, but he likens himself to an explorer venturing into a new land. “I am really going out there and looking at this wild universe and finding things that I hope will eventually lead to proofs,” he says. He is just now beginning to analyze his mammoth cache of zeros. “We simply don’t know what surprises the data might hold,” he declares. Odlyzko notes that this explorational view of mathematics is very much part of the subject’s tradition. Indeed, a marvelous new book by Amir Alexander, Geometric Landscapes (Stanford University Press), traces the history of the idea of the mathematician-explorer and shows how the rhetoric of discovery was integral to the way in which mathematicians of the scientific revolution conceived of themselves and their work.
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.” After so many years on the cliff face, no one has a greater investment in the problem than Odlyzko. I ask him if he thinks it will be resolved in his lifetime. Before answering, he pauses and on the other end of the phone I can hear a slow intake of breath. Yet his answer, when it comes, is full of optimism: “For all we know, it may have been done yesterday,” Odlyzko says. “It may be done tomorrow.”
Then again, he adds, “It may take another hundred years.”
