April 19th, 2017

The Riemann Zeta Function is defined as:

$$\zeta(s) = \sum\limits^\infty_{n=1}\frac{1}{n^s}$$

It looks really scary, but the big $\Sigma$ just means you add a bunch of stuff. So it could be written:

$$\zeta(s) = \frac{1}{1^s} +  \frac{1}{2^s} +  \frac{1}{3^s} +  \frac{1}{4^s} + ...$$

So, all you do is put in a value for $s$ and you get some output $\zeta$. For example, if you put in $s = 2$ you get:

$$\zeta(2) = \frac{1}{1^2} +  \frac{1}{2^2} +  \frac{1}{3^2} +  \frac{1}{4^2} + ...$$

Which equals:

$$\zeta(2) = \frac{1}{1} +  \frac{1}{4} +  \frac{1}{9} +  \frac{1}{16} + ...$$
And we can calculate the value of each fraction:

$$\zeta(2) = 1 + 0.25 +  0.111 +  0.0625 + ... \approx 1.645$$

So we can say that the Reimann Zeta Function of 2 is about 1.645. See? Not so bad. It turns out if you keep plugging in numbers into this function (including complex numbers, which are like number pairs) you get some very beautiful patterns (where the colours are a way of representing these complex pairs of numbers). There is a very special part of this function called the 'critical line', and it seems like any number on this line that, when put into the Reimann Zeta Function, gives $\zeta = 0$ tells us how to find a Prime Number (prime numbers, to date, can only really be found by trial and error). Prime numbers are incredibly valuable in many aspects of science, computers, and math. As such, if this property of the critical line is prove to indeed be true, and not just a lucky coincidence, then there is a one million dollar prize waiting for the brain that accomplishes the feat. An excellent video by youtuber 3blue1brown explains this much better than the above attempt.

-E