The **Zeta function** ζ(s) is defined as above, for example :

ζ(2) = 1/1^{2} + 1/2^{2} + 1/3^{2} … = π^{2} / 6.

This converges for s>1 (i.e. tends towards a limit), but diverges for anything else. The *Riemann *Zeta function extends this range to allow us to compute the result for any complex number. The **Riemann Hypothesis** claims that the ‘zeros’ or ‘roots’ of this extended function, i.e. solutions *s *for which ζ(s) = 0, have the form 1/2 + ai, i.e. where the real part of the complex number is always 1/2 (as well as certain ‘trivial’ roots which don’t have this form).

**Unsolved! **Proposed by Bernhard Riemann in 1859. This is one of the 7 ‘Millennium Prize’ problems, **for which there is a $1m reward**.

Get cracking!

There’s a number of consequences if this is true, but perhaps the most important is that it reveals the *distribution* of the prime numbers. The *Prime Number Theorem* allowed us to estimate the number of primes up to a given number. If armed with all the roots of the Riemann Zeta function, then we can work out the exact number!