So how do you sum — or, if you prefer, associate a value with — ?
One way is zeta function regularization. For this we start with the sum
For example, . This series converges to the limit . In fact converges for any complex where the real part .
For , and this diverges. If you approach along the real axis you find increases without limit. Off the real axis, things are a little different. At , for example, the sum fails to converge, but as you approach t from the right, approaches . Similar behavior is found elsewhere on the line, other than .
That suggests there might be a way to construct a function that is equal to for but which has well defined values elsewhere, except . And indeed there is: analytic continuation.
Imagine I give you the following function: for real . Outside that interval is undefined. But you obviously could define another function which is defined on the whole real number line and has the property that in the range where is defined. Obviously is continuous, and is differentiable everywhere.
On the other hand, you could instead define as being quadratics grafted onto the line from to :
which has the same properties. Or you could use cubics, or quartics, or, well, anything provided it has the right value and derivative at and . There’s an infinite number of ways to continue to the entire real line.
In the complex plane you can do something similar. I give you a function defined for within some region of the complex plane. is analytic, that is, it has a complex derivative everywhere it’s defined. Then you can give me an analytic function defined everywhere in the complex plane and equal to everywhere is defined. (I’m being sloppy and informal here; there could be poles where neither function is defined, for example.)
Here’s the thing, though: Unlike on the real line, is unique. There is exactly one analytic function that continues my analytic function to the entire complex plane.
So, getting back to our sum (which is analytic), we can define an analytic function for , whose behavior for is given by analytic continuation. One can show
where is the usual gamma function. has a pole at but is well defined everywhere else. is known as the Riemann zeta function.
Now, we know is the value of wherever that sum converges. Zeta regularization just assigns the value of to that sum where it does not converge as well. For instance, when , we have , and .
The somewhat notorious sum of the positive integers, , is , to which is assigned the value . If you want to start an argument on the Internet, claiming that is a good way to do it. Of course that claim glosses over a lot.
It turns out the negative even integers are (“trivial”) zeros of the zeta function, so by this summation method. Generally, for integer exponents,
where is the nth Bernoulli number,
and on from there.