The Music of the Primes: Why an unsolved problem in mathematics matters

At first sight Euler’s product doesn’t look as if it will help us in our quest to understand prime numbers. After all, it’s just another way of expressing something the Greeks knew more than two thousand years ago. Indeed, Euler himself would not grasp the full significance of his rewriting of this property of the primes.

The significance of Euler’s product took another hundred years, and the insight of Dirichlet and Riemann, to be recognised. By turning this Greek gem and staring at it from a nineteenth-century perspective, there emerged a new mathematical horizon that the Greeks could never have imagined. In Berlin, Dirichlet was intrigued by the way Euler had used the zeta function to express an important property of prime numbers – one that the Greeks had proved two thousand years before. When Euler input the number 1 into the zeta function, the output

spiralled off to infinity. Euler saw that the output could spiral off to infinity only if there were infinitely many prime numbers. The key to this realisation was Euler’s product, which connected the zeta function and the primes. Although the Greeks had proved centuries before that there were infinitely many primes, Euler’s novel proof incorporated concepts completely different to those used by Euclid.

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