7, 11 and so on, the numbers only divisible by 1 and themselves. Using primes,
we can form the expression
and this turns out to be another way of writing ξ(s), the Riemann zeta
function. This tells us that knowledge of the Riemann zeta function will throw
light on the distribution of prime numbers and enhance our understanding of the
basic building blocks of mathematics.
In 1900, David Hilbert set out his famous 23 problems for mathematicians to
solve. He said of his eighth problem, ‘if I were to awaken after having slept for
five hundred years, my first question would be: Has the Riemann hypothesis
been proven?’
Hardy used the Riemann hypothesis as insurance when crossing the North sea
after his summer visit to his friend Harald Bohr in Denmark. Before leaving port
he would send his friend a postcard with the claim that he had just proved the
Riemann hypothesis. It was a clever each way bet. If the boat sank he would
have the posthumous honour of solving the great problem. On the other hand, if
God did exist he would not let an atheist like Hardy have that honour and would
therefore prevent the boat from sinking.
The person who can rigorously resolve the issue will win a prize of a million
dollars offered by the Clay Mathematics Institute. But money is not the driving
force – most mathematicians would settle for achieving the result and a very high
position in the pantheon of great mathematicians.