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.
The sum as you've defined it only converges on the right-half plane Re(s)>1.
Not terribly exciting, most of the interesting action happens for complex values of s
The most interesting feature of the zeta function (the behavior near s=1) does occur on the real line.
You're right about the convergence of the sum, but of course it's pretty easy to analytically extend it to all complex s (except s = 1, of course). But generally the behavior there is not so theoretically important compared to its behavior along the line s = 1/2 + iy, for real y, which is where the nontrivial zeros of Riemann hypothesis fame live.
Matt, where the most interesting behavior is occuring is somewhat subjective. Note that the prime number theorem itself is equivalent to the claim that there are no zeros on the line Re s=1. And Chebyshev's bounds can be derived by close examination of the singularity at s=1. And the integer values of zeta have very interesting connections to the Bernoulli numbers and a lot of deep topics. If one primarily cares cares about the error bound of the Prime Number Theorem then yes, the 1/2 line is probably the most interesting part although also note that right now, we can't even prove that there aren't zeros arbitrarily close to the Re s =1 line. So if one cares about that error, the interesting area is the strip with 0< Re s <1.
All right, how about this: the most interesting point is the first zero discovered in the critical strip not on the line 1/2 + iy. ;)
Gah, bad interpretation of rest of comment by HTML cut off the remainder. Last part of sentence rephrased should end " the interesting area is the strip with real part between 0 and 1".
The significance of Euler's Identity is that the prime
numbers appear on just one side of the equation but not
on the othet. On the summation side all the irregularity
and idiosyncracy of the prime numbers have vanished and we
have something completely regular. But all the prime numbers
appear on the other side.
Everything about the Riemann Zeta Function is endlessly
fascinating. As a start Euler's Identity shows that if
the number of primes were finite then pi^2 would be rational (from the special value at s=2). Since it can
be shown that pi^2 is irrational ( original proof
due to Lambert and Legendre - short proof due to Hermite)
we get the funiest proof of the infinitude of the primes.
This is not just a joke. The key is that to prove that
zeta(2) is irrational requires only working with the left
side of Euler's Identity and requires knowning nothing
about prime numbers. But by Euler's Identity this fact
implies something about the primes (namely they cannot be finite in number). In general we can establish results
about the Riemann Zeta Function based on the left hand
formula requiring no information on the primes and then Euler's Identity ( and of course the Explicit Furmulas)
give us information about the primes.
Then information about the primes can pass from the right side of the identity to get information about the Riemann
Zeta Function which information suitably processed analytically feeds back to the right side (and also through
the Explicit Formulas) to get more refined information about the primes.
HOW NOSTRADAMUS WON ALL THE PARANORMAL PRIZES!
THE HIGH PRICE OF REVOLUTION
OK, here's an issue I've had with regard to the usual discussion of the zeta function. I'm not sure I can well articulate it, or that it is necessarily even coherent. But perhaps those wiser than I can address it, and show me the error of my ways.
One thing that has struck me about the Zeta function and its relation to prime numbers is that the definition of the Zeta function does involve the use of an infinite series. While the infinite series does not directly involve the prime numbers, it's pretty apparent that the terms of that series can interact in such a way that, as the series gets amplified at each step, the newest step can in certain respects reverse direction of the overall effect up to that step (that's one way of understanding the zero crossings).
The point is, buried within its definition is a mechanism whereby a countably discrete number of things occur. It is not terribly surprising, perhaps, that an equivalence of this to the behavior of the primes might be found (or at least that such an equivalence might be found for SOME such function, if not the zeta function itself.)
Now if the zeta function in its definition did NOT involve an infinite series -- if, for example, it could be shown to be the exact equivalent of a function with no such appeal to an infinite series -- then it would seem quite remarkable indeed that it should have such a relation to prime numbers.
But does such an alternative definition exist? I guess I haven't heard of any such.
Maybe one way of understanding my concern about the zeta function is that, because of the way it is defined, and the way its distinct terms interact, it more or less has a full description of the integers built into it.
And where there is the behavior of integers, there is the behavior of the primes.