A beautiful introduction to the Holy Grail of mathematics
Two years back, exactly to this day, I had visited Alliance Francaise in Delhi with a friend. They were going to add new titles to their library, so some part of the older collection had been put up for sale. As expected, they had books and editions not usually found in bookshops or online bookstores. Amongst the gems I picked up that day was Marcus du Sautoy’s The Music of the Primes. In all honesty, I purchased the book because it was a hardcover with an intact dust jacket, it was on mathematics, the book was in perfect condition and didn’t have a single pen or pencil mark.
The book would turn out to have two interesting connections with my earlier forays into the world of maths, and I was oblivious to both at that time.
The first one was the author of the book. A few years earlier I had watched a BBC Four series titled The Story of Maths. It was presented by Sautoy and although I hadn’t gathered his name, his face had stuck.
The second was what the book was about – the Riemann Hypothesis, the most important unsolved problem in mathematics.
The importance of the hypothesis can be gauged from the fact that it is the only problem that occurs on both Hilbert’s list of twenty three problems and the seven Millennium Prize problems – the two most important lists of unsolved problems that have come up over the last century and a quarter and which have provided an impetus and given a general direction to mathematical research. The former were presented by David Hilbert at the International Congress of Mathematicians at Paris in 1900, and the latter were put forth by the Clay Mathematics Institute in the year 2000.
The Riemann Hypothesis was put forward by Bernhard Riemann, one of the most important mathematicians of all time, in the year 1859. Riemann brought about a shift in perspective in the philosophy of mathematical research. He believed it was more important to understand the hidden structure of maths than to try to solve specific questions. In that sense, he heralded a revolution in the psychology of approaching mathematical problems, a culture that continues to this day. In fact, half a century later, Einstein would discover that Riemann’s new mathematical language, of which the Hypothesis was merely an incidental observation, was perfectly suited to express his transformative ideas of special and general relativity.
Over time, hundreds of other results have come up which proceed by assuming either the truth or falsity of the Riemann Hypothesis. Thus the resolution of this hypothesis, either way, will have huge implications for the mathematical edifice.
At its heart, the Riemann Hypothesis asks a simple question – is there some hidden pattern in the distribution of prime numbers?
Despite being the building blocks of arithmetic, prime numbers are really not that well understood. Why does their distribution seem so random? Are they following some pattern? Is there some logical structure that permeates them, and which could be used to catch a glimpse about their mysterious world? Given a prime number, how long do we have to count upwards till we encounter another one?
Mankind’s quest to understand prime numbers actually has a pretty rich history that goes back over two thousand years to Euclid who, in his book Elements published around 300 BCE, provided a very simple argument to prove that there are an infinite number of primes. Over time a number of mathematicians have worked on it. In fact, the list of mathematicians whose work has either directly or indirectly helped in our understanding of primes sounds like a who’s who of the history of mathematics – Euler, Fermat, Gauss, Dirichlet, Fourier, Hilbert, Riemann, Ramanujan, Hardy, Gödel and Turing.
The modern history of the story, as well as this book, starts with Gauss who brought about the first fundamental shift in how we think about prime numbers. Instead of asking when the next prime number will occur, he asked, instead, how many primes occur up to any given number ‘n’. John Napier had come up with his logarithm tables just a few decades before, and Gauss realised he could use them to convert multiplications of huge numbers into simple addition. He developed his ideas and came up with his path-breaking approximation of n/log(n).
It was Riemann, however, who shifted gears and brought about the second fundamental shift by opening up a new landscape to understand the problem. He transformed the puzzle of the distribution of primes into the properties of a certain curve in three dimensions. The Riemann Hypothesis essentially states that the set of coordinates where the height of the curve is 0, follow a particular pattern.
The Music of the Primes provides an extremely enriching and exhilarating vision of the developments in the study of primes over the last two hundred odd years when, really, most of the progress has been made. Among the many interesting things I came to know, two have really stood out.
The first one was realising that in 1976 a group of mathematicians, working on a theorem put forth by the Russian mathematician Yuri Matiyasevich, came up with a formula in 26 variables using which it is possible to generate all the prime numbers. The second was realising that even though we have such a formula, it is not as valued today because the focus has now shifted following Riemann’s gear change, and it was precisely this striving to make sense of the hidden structure and behaviour of mathematics that has led to connections between, lo and behold, prime numbers, quantum theory and chaos theory. That’s right. Please do yourself the favour of reading the previous sentence again, and then kindly proceed to pick up your jaw that may have fallen to the floor.
Sautoy is the Simonyi Professor for the Public Understanding of Science at the University of Oxford, a chair that was created in 1995 and was occupied by Richard Dawkins till 2008, when Sautoy took over. His choice for the title of the book reflects his ability to realise that referencing the Hypothesis would end up restricting his audience to a very niche subset of mathematical enthusiasts, whereas a title such as The Music of the Primes is at once both mysterious and evocative, and would be able to vibe even with the general public.