Stumbling through mathematics: the Riemann hypothesis

The Riemann zeta function ζ(s), where s is a complex variable, is given by:

Leonhard Euler (1707-1783) discovered that ζ(s) could be represented as an infinite product involving the prime numbers:

where p represents the prime numbers 2, 3, 5, ... The "trivial" zeros of the zeta function are ζ(-2) = 0, ζ(-4) = 0, ζ(-6) = 0, etc. The following is a Maple plot of a few of the trivial zeros:

The Riemann hypothesis states that all of the non-trivial zeros of the Riemann zeta function line on the line Re(s) = 1/2. Below is a plot of |ζ(1/2 + iy)| which shows a few of the non-trivial zeros:

As of 2004 over 10,000,000,000,000 non-trivial zeros have been calculated. All of them lie on the line Re(s) = 1/2; however, this does not constitute proof of the Riemann hypothesis. To date no one has been able to prove (or disprove) the Riemann hypothesis. A $1M prize has been offered to anyone who offers a valid proof one way or the other.

An excellent account of the Riemann hypothesis and its history is Prime Obsession, by John Derbyshire.