2006-02-04 by
qntm

The Riemann Hypothesis is a problem in mathematics which is currently unsolved.

To explain it to you I will have to lay some groundwork.

First: complex numbers, explained. You may have heard the question asked, "what is the square root of minus one?" Well, maths has an answer and we call it i. i multiplied by i equals -1. If the real number line ...-4, -3, -2, -1, 0, 1, 2, 3, 4... is represented as a horizontal line, then the numbers ...-4i, -3i, -2i, -i, 0, i, 2i, 3i, 4i... can be thought of as the *vertical* axis on this diagram. The whole plane taken together is then called the complex plane. This is a two-dimensional set of numbers.

Every complex number can be represented in the form `a`+`b`i. For real numbers, we simply take `b`=0.

Next: functions. In mathematics, a function is a black box which, when you put a number into it, spits a different number out. A function is represented by a letter - usually "`f`". If you put a number `x` into the function you call `f`, then what `f` then spits out is written "`f`(`x`)".

In most cases there is a convenient way to express `f`(`x`) in terms of `x`. For example, `f`(`x`)=`x`^{2} is a very simple function. Whatever `x` you put in, you'll get `x`^{2} out. `f`(1)=1. `f`(2)=4. `f`(3)=9. And so on.

You're probably most familiar with real functions, or functions where you put a real number in and always get a real number out. HOWEVER. There's nothing stopping you from putting these weird new complex numbers into a function. For example, if `f`(`x`)=`x`^{2} and we let `x`=i, which is the square root of minus one I mentioned above, then you'll get `f`(i)=-1. That's just the beginning of what's more generally known as complex functions - where you can put any complex number `a`+`b`i in and get (potentially) any complex number out.

The Riemann Zeta Function is just such a complex function. "Zeta" is a Greek letter which is written "ζ". For any complex number `a`+`b`i, `ζ`(`a`+`b`i) will be another complex number, `c`+`d`i.

The actual description of the Zeta Function is too boringly complicated to explain here.

Now, a zero of a function is (pretty obviously) a point `a`+`b`i where `f`(`a`+`b`i)=0. If `f`(`x`)=`x`^{2} then the only zero is obviously at 0, where `f`(0)=0. For the Riemann Zeta Function this is more complicated. It basically has two types of zeros: the "trivial" zeroes, that occur at all negative even integers, that is, -2, -4, -6, -8... and the "nontrivial" zeroes, which are all the OTHER ones.

As far as we know, *all* the nontrivial zeroes occur at 1/2 + `b`i for *some* `b`. No others have been found in a lot of looking... but are they ALL like that? The Riemann Hypothesis suggests that they are... but nobody has yet been able to prove it.