## The Riemann Hypothesis – Karl Sabbagh

March 11, 2007

The Riemann Hypothesis
The Greatest Unsolved Problem in Mathematics
Karl Sabbagh (Farrar Strauss Giroux, 2003)

Posted 11 March 2007.

It has now been more than 150 years since Bernhard Riemann, the great German geometer and analyst, set forth his famous hypothesis, and efforts to prove (or disprove) it have occupied many of the finest mathematical minds of each succeeding generation. The hypothesis is tantalizing for many reasons, but perhaps chief among them is that, if true, it would yield deep insight into the nature of prime numbers. More precisely, the zeros of the complex-valued Riemann zeta function $\zeta(z)$ could, if known, be used to calculate the distribution of the prime numbers along the number line. The Riemann hypothesis states that the zeros of this function can all be written in the form $z = 1/2 + i b$. That is, they all lie on the so-called critical line defined by $Re(z) = 1/2$.

A proof of this seemingly simple statement has turned out to be perniciously difficult to produce, and Sabbagh’s book is partly an account of the history of attempts to do so, and partly an introduction to contemporary mathematicians whose work circles around the problem. In this sense it is an interesting view into what motivates mathematicians, and Sabbagh has some intriguing things to say about the art of mathematical thinking. He also makes clear that though the Hypothesis itself has resisted all efforts at proof, the attempted proofs (and there have been many) have greatly enriched mathematics in the meantime. Connections to the Riemann hypothesis have been found in number theory, of course, but also in operator analysis, geometry, and even quantum theory.

Sabbagh is himself a non-mathematician, and while this might make him just the man to explain the topic to non-mathematical readers, it does make the book a little odd for someone with a decent mathematics education. He wanders among the mathematicians like a stunned explorer among the natives, marvelling at their strange thoughts, their impenetrable language, and their peculiar sense of humour. This tone of astonishment adds considerable charm to the narrative, but it also means that when the subject matter becomes technical the writing begins to grasp at analogies and metaphors, and one struggles to ascertain what he’s really trying to say.

I was able, however, to glean a few hard technical facts. I learned, for instance, that the first ten trillion zeros of the Riemann zeta function have been checked by computer and they all lie on the critical line. This might be enough to convince one that the Riemann hypothesis is true, but cases have been known in the past where even such apparently uniform behaviour has been shown to change when pushed further. And besides, that kind of empirical verification doesn’t satisfy a truly mathematical mind. More formal methods have been able to establish a number of interesting results. For instance, all of the zeros of the Riemann zeta function are known to lie in the critical strip $0 < Re(z) < 1$, and it has been proven that at least forty percent of the zeros fall on the critical line. Furthermore, if there is a zero away from the critical line it will be paired with a second, the two arranged symmetrically around the critical line. All of which is very interesting.

[A mathematician speaks]
Someone who had begun to read geometry with Euclid, when he had learned the first proposition, asked Euclid, “But what shall I get by learning these things?” whereupon Euclid called his slave and said, “Give him three-pence, since he must make gain out of what he learns.”