October 2003

David Harris


The Music of the Primes by Marcus du Sautoy

In 1900, the great German mathematician David Hilbert presented a lecture to the International Congress of Mathematicians in which he outlined a program for the development of mathematics. His program was defined by a set of 23 unsolved mathematical problems that represented the most significant gaps in mathematical knowledge at the time.

A century later, only one of the problems remains completely unsolved. The problem appeared in a 10-page paper by Bernhard Riemann, published in 1859. The Riemann Hypothesis, as it is now known, makes claims about the nature and distribution of prime numbers, the building blocks of mathematics. It has fundamental importance to many areas of mathematics but also impacts on the everyday world. It is interesting and important enough that there is $1 million bounty for its solution, offered by the Clay Mathematics Institute.

Information security currently rests on the apparent difficulty of factoring large numbers into primes. If the Riemann Hypothesis is proven, the proof may provide a quick way to factor numbers, and hence crack all of the major cryptographic systems currently in use. For this reason, groups ranging from banks to national security agencies are investing heavily in the abstract and abstruse pure mathematics behind the Riemann Hypothesis.

The Riemann Hypothesis has never gained as much public attention as other unsolved mathematical problems, presumably because it is not only abstract, but also difficult to state concisely. In contrast, Fermat's Last Theorem is familiar to a surprisingly large segment of the population, well beyond professional mathematicians. And when Andrew Wiles first claimed proof of the Theorem in 1993 (I still have the front page of the New York Times that announces the proof), it was common for abstract mathematics to suddenly appear in conversations at dinner parties.

Although Fermat's Last Theorem can be stated in a few sentences*, the Riemann Hypothesis is not even defined in The Music of the Primes until page 99. This is partly because it is reasonably complex, but also because Music is far more than just a discussion of the Hypothesis. In fact, Music is a wide-ranging historical survey of a large chunk of mathematics with the Riemann Hypothesis acting as a thread tying everything together. The scope of the book is sufficiently wide that I am sure that even most mathematicians will learn a lot about the history of their field, and many will be surprised that a true understanding of work on the Riemann Hypothesis also requires knowledge of quantum physics.

This connecting of the fields of mathematics and physics is not merely a ploy on the part of the author to be interesting -- the nature of the Riemann Hypothesis requires it. The excursions into physics may seem controversial to some mathematicians but perhaps even more controversial is du Sautoy's claim, "mathematicians admit that it could well be a physicist who finally proves the Riemann Hypothesis." That would really piss them off. I know this because it creates feelings of intense anger for the mathematician in me but extreme delight for my physicist side.

Regardless of your background, if you have any curiosity about what the world of mathematics is really about (as opposed to how it seemed at school), this book may be the introduction for you. Some of the maths are tough but the history and storytelling paints a convincing (and appealing) picture of the world of professional mathematics.

There is a flaw in the approach of this book. Du Sautoy never really explains why primes are so important and why we should really care. That we should has become so ingrained in the minds of professional mathematicians that they often forget that the answer is not so obvious to others. He also skims over so many topics using superlatives and appeals to emotion that it is easy to miss that you never really found out what was going on. You'll get to the end of the book feeling that you really learned something but when it comes down to it, or if you need to explain the concepts to somebody else, you'll find that the illusion was greater than what you actually learned. So don't count on too much, just enjoy the ride.

Tackling the details of the Riemann Hypothesis may be a little frightening if you haven't played with maths since school but try out my explanation below**. Yes, it is tough, but you never know -- you might just find it's palatable. If you don't want to tackle the details, just imagine that there is a particular mathematical landscape that occasionally drops down to sea level. The Riemann Hypothesis just says that all of the sea level points must lie in a particular straight north-south line.

This seems to have absolutely nothing to do with anything at first glance. However, Riemann showed that there is an intimate connection between his mathematical landscape and the prime numbers. Prime numbers are those that can't be divided by any number (except themselves and one). You can build any number at all out of the primes, but the primes are not predictable. The first few are 2,3,5,7,11,13,17,19,23... If you write out all the numbers and circle the primes, they appear to be dispersed at random. One of the big challenges of 19th century mathematics was to see if there was some hidden order to the primes. The great mathematician Gauss found something that looked like a pattern -- he created a formula that said how often primes should occur on average -- but he couldn't prove he was right.

This is where Riemann's mathematical landscape comes in. By knowing the height of the landscape, Riemann could beat Gauss's guess and predict exactly how many prime numbers there were up to any given number. For example, there are 25 prime numbers less than 100, 168 primes less than 1000, and 50,847,534 primes less than one billion. Riemann's formula gets these numbers exactly right.

Riemann's landscape has some very special properties and one part of the landscape constrains what happens elsewhere. Each sea-level point corresponds to various waves in the landscape and these waves all add up, just like water waves or even sound waves. This is where the analogy with music appears. Dissecting the landscape into its constituent waves is like breaking down an instrument's sound into its fundamental and harmonic frequencies. But in the mathematical landscape we find the frequencies correspond to primes. These waves also appear in another form in quantum physics, where every particle also has wavelike properties.

Music of the Primes is right up-to-date covering some of the latest main developments in the quest to prove Riemann's Hypothesis. Many of the major mathematicians of the 19th and 20th century make an appearance and, although I haven't counted, it seems like a majority of Field's Medal winners (Nobel Laureate equivalents). This is not a happy story, in that the Hypothesis is not solved by the end. There is tragedy -- mathematicians literally went mad in their quests -- but also much joy.

Many milestones along the path to Mount Riemann seem to have been reached, but nobody is even sure that the path goes all the way. Perhaps the Riemann Hypothesis is one of the mathematical statements that is true but has no proof. I'd explain that further but you'll find out what that means in the chapter on Kurt Godel. What we do know is that the search for a proof has spawned entire new fields of mathematics, and provided solid foundations for others. The adventure is as exhilarating as any purely intellectual endeavor can be, and I am sure that if the Riemann Hypothesis is proven, many mathematicians will feel a great loss, until they get to work on the Clay Millennium problems...

The Music of the Primes by Marcus du Sautoy
Harper Collins
ISBN: 0066210704
352 pages

Further reading:

A mathematical definition of the Riemann Hypothesis

Clay Mathematics Institute Riemann Hypothesis resources

Play with the Riemann Hypothesis in a java applet

(*) Fermat's Last Theorem states that there are no solutions in whole numbers, x, y and z, for the equation x^n + y^n = z^n when n>2. When n=2, this equation does have solutions - it is Pythagoras' Theorem (x^2 + y^2 = z^2), which may be familiar from elementary school mathematics relating the lengths of the sides of a right-angled triangle.

(**) We start with something called the zeta function. This is a formula into which you plug a number, and you get one out. However, the formula has an infinite number of parts to it so it is not even obvious if it has a well-defined answer in all cases. To evaluate the zeta function for a number x, you add up 1/(1^x) + 1/(2^x) + 1/(3^x) +... (I think you get the picture here - you divide 1 by the xth power of each number 1, 2, 3, etc. and add them all up.)

For example, if x=1, then the zeta function z(1)=1 + 1/2 + 1/3 + 1/4 + 1/5 +... It turns out that this is infinite so it doesn't tell us a great deal. Let's try another: x=2. Then z(2) = 1 + 1/4 + 1/9 + 1/16 + ... Now this does have a finite solution. Somewhat surprisingly, this infinite sum adds up to the value pi^2/6. (pi is the number 3.14159... that you know and love from high school geometry.)

So we know that sometimes the value of the zeta function might be infinite, other times it might be finite. The question Riemann asked is when does the zeta function have the value zero. i.e. when do all the infinite number of terms perfectly cancel each other out. If you just plug in ordinary numbers, you'll never get zero - the answer is always positive - but if you consider imaginary or complex numbers, everything changes. Imaginary numbers are numbers that when squared give a negative number. This sort of number is not something you deal with in everyday life, even if the numbers on your credit card bill might seem imaginary. It even takes a new notation to write them down. Rather than going into the details, I'll have to ask you to trust me that all of modern physics is founded on imaginary numbers and that with a small amount of practice, you can manipulate them just as easily as real numbers. They are often best thought of as arrows lying on a page rather than numbers on a line.

So what does the Riemann Hypothesis say? It says that the only places where the zeta function is zero have a real part of 1/2 and some non-zero imaginary part. In other words, if you plotted a landscape of real numbers going east-west and imaginary numbers going north-south, all the zeroes of the zeta function would lie on a straight north-south line that passes through the real number 1/2.

I told you it was pretty abstract.