The zeta function is a nineteenth century concept exploited by the German mathematician Bernhard Riemann to reveal many of the secrets of the primes. Prime numbers have fascinated mathematicians ever since the ancient Greeks discovered they are the building blocks of all numbers. But they also represent one of the greatest enigmas in our subject. The zeta function has given us a glimpse of what makes prime numbers tick. The zeta function is like a magical looking glass. On one side of the mirror sit the primes, a chaotic bunch of numbers with no rhyme or reason to them; on the other side of Riemann’s mirror, the primes are transformed into geometry, something completely different where patterns and order prevail.
Since Riemann’s use of the zeta function, many variants have been exploited to reveal extraordinary connections between different bits of the mathematical world. In my own research, the zeta function revealed a strange connection between the world of symmetry and the theory of elliptic curves. Without the zeta function I would never have know these two things were related.
Marcus Du Sautoy is professor of mathematics at Wadham College, Oxford, senior media fellow at the Engineering and Physical Sciences Research Council, and presenter of the BBC4 programme Mind Games. Author, The Music of the Primes (Harper Perennial), which tells the story of the zeta function (buy this book from Amazon(UK)).