The Music of the Primes: Why an unsolved problem in mathematics matters. Marcus Sautoy du

the first sign that the zeta function might reveal unexpected links between seemingly disparate parts of the mathematical canon. The second strange connection that Euler discovered was with an even more unpredictable sequence of numbers.

      Rewriting the Greek story of the primes

      Prime numbers suddenly enter Euler’s story as he tried to put his rickety analysis of the expression for on a sound mathematical footing. As he played with the infinite sums he recalled the Greek discovery that every number can be built from multiplying prime numbers together, and realised that there was an alternative way to write the zeta function. He spotted that every term in the harmonic series, for example , could be dissected using the knowledge that every number is built from its prime building blocks. So he wrote

      Instead of writing the harmonic series as an infinite addition of all the fractions, Euler could take just fractions built from single primes, like , and multiply them together. His expression, known today as Euler’s product, connected the worlds of addition and multiplication. The zeta function appeared on one side of the new equation and the primes on the other. In one equation was encapsulated the fact that every number can be built by multiplying together prime numbers:

      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.