The Zeta function ζ(s) is defined as above, for example :
ζ(2) = 1/12 + 1/22 + 1/32 … = π2 / 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.
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!