A rehash for all of you who get bleeding brains when I go into math mode:
There is a function (the zeta function) that predicts the distribution of the prime numbers. In 1859 this pretty neat mathematician named Riemann presented a short paper in which he conjectured that the non-trivial roots of the zeta function would all have a real part = 1/2.
Lost you, didn't I? Okay, go back to high school algebra and those equations you had to solve. The ones that looked like x2 + 3x - 4 = 0. You'd do some arithmetic and find out that you could rewrite the equation as (x + 4)(x - 1) = 0 and the solution was x = -4 or x = 1. Those were, more or less, the trivial roots.
Although it's more complicated (I'm guilty of major simplification in the previous paragraph), the trivial roots of the function in the Riemann Hypothesis are all negative numbers (-2, -4, -6, .....) and the non-trivial roots are all complex numbers. The hypothesis says that the non-trivial roots will all be 1/2 + yi (y is any real number and i is the square root of -1).
Fast forward to 2007 and nobody has proved or disproved the hypothesis. It's a really BIG deal for a whole lot of reasons. A draft paper has appeared on ArXiv claiming to disprove the hypothesis. Keep in mind that this is a draft and has not been peer-reviewed in any manner. Nevertheless, it's getting attention. Check the entries on Gooseania and Ars Mathematica.
Even though my interest in the zeta function has to do with that real part "y" of the non-trivial roots, I've downloaded the paper. I can tell you that 8 pages into it, I'm cross-eyed. LOL