In pure mathematics there’s not too many function studied more than the Riemann zeta function. For reasons of historical tradition, the generic variable name that’s usually used is s instead of z. (The function is mostly interesting in terms of complex analysis, so x would be a bit unorthodox too.) It’s defined in the following way:
On the real line, it looks like this:
Not terribly exciting, most of the interesting action happens for complex values of s. The reason that the action is interesting is that the zeta function is intimately connected to the prime numbers, despite there being no obvious connection between the prime numbers and the definition I posted above. But there is such a connection, and it’s one of the most important connections in pure mathematics. To make this connection, I’ll follow the presentation of the argument from John Derbyshire’s excellent pop-math book Prime Obsession. The argument itself is originally due to Euler. I’ll also drop the summation notation until the very end to save space. So start off with the zeta function we wrote down above and multiply both sides by 1/(2^s). We’ll get this:
Now take that and subtract it from the original definition of the zeta function. We get:
Because we subtracted off all the even-numbered terms. Now repeat the process, except starting off by muptiplying by 1/(3^s). You’ll end up getting rid of all the terms that are multiples of 3:
We’ve gotten rid of the multipes of 2 and of 3. However, we’ve also gotten rid of all the multiples of 4, because all the multiples of 4 were also multiples of 2. So the next thing to get rid of is the multiples of 5. (which also gets rid of the 10s and 15s and 20s, etc). Then the 6 (and its multiples) are gone because we already got rid of the 3s. So now the next thing to get rid of is the 7s. In other words, each time we multiply by and subtract, we’re eliminating the next lowest prime.
But look at the right hand side of the equation – each time we repeat our multiply-and-subtract procedure the remaining leading term (after the 1) gets smaller and smaller. Eventually they’ll be smaller than any number we care to name. At that point only the 1 will be left over, so after we’ve repeated our procedure an infinite number of times we have:
So divide out the zeta function and substitute in its definition, and you have this rather astonishing relationship:
Where the p’s are prime numbers. This is a pretty astonishing fact – the prime numbers are deeply encoded in the zeta function. Actually the relationship goes considerably deeper, though that exploration would take us quite a but far afield.
As a check, let’s take s = 2 and add the first 100 terms of the sum and the first 100 terms of the products. They happen to be about 1.63498 and 1.64452 respectively, which is pretty good given the fairly small number of terms we’ve used. Using more terms would make the relationship as exact as we want.
Not only is the zeta function’s relationship to the primes an interesting thing to study, it might be a profitable one. A conjecture by Riemann about the zeta function is probably the most important open question in mathematics, and there’s a million buck reward for its solution.