There is a popular mathematical result which says that the sum of all natural numbers is -1/12. I have discussed it in detail here. This looks very unintuitive to a first time observer. In fact, most people would say that this is some kind of mathematical trickery. How can a bunch of positive numbers sum up to a negative fraction, right? Actually, there is a very real purpose to this whole thing of adding up all the natural numbers to get a negative fraction as the result. However, our general sense tells us that this shouldn’t be possible. The discussion in one of my previous blog posts was about the mathematics involved in this result. This discussion is more about the underlying fundamentals and where these results come from. So how do we explain this situation? Where is it used in real life?
First of all, what is “zeta function”?
The zeta function (also called the Reimann Zeta Function) is defined as shown below:
ζ(s) = 1 + 1/2s + 1/3s + 1/4s + ...
If ‘s’ is greater than one, then the each successive term gets smaller and smaller quickly, and the sum starts converging. We can add them all up to get a finite value. If ‘s’ is less than or equal to 1, this infinite sum doesn’t converge. As in, as we keep adding more terms, the sum starts diverging. This makes it impossible for us to add them up in any reasonable way. It just ends up approaching infinity! To put another way, we say that the sum is not properly defined. But as it so happens, mathematicians don’t like it when something is not “properly defined”. So they came up with a formulation where ζ(s) doesn’t go to infinity for values of ‘s’ less than 1!
How do we prevent ζ(s) from going to infinity?
This is where complex analysis comes into picture. Analysis is the type of mathematics that deals with functions, infinite series, and the related calculus. It’s often contrasted with algebra, which usually considers mathematical concepts that are discrete rather than smooth. This is a huge oversimplification of a rather intricate mathematical concept, but it will suffice for now. It’s called “complex” because it deals with complex numbers, and not because it’s actually complex. Complex analysis just deals with functions that are pertaining to complex numbers.
An interesting thing to note about complex analysis is that if a function of complex numbers is sufficiently smooth, then it is very highly constrained. Wait, what does that mean? It means that if we have a function that is smooth over a small region, then we can estimate its behavior elsewhere. To put a mathematical spin on it, we can say that if a function is analytic and we know how a function behaves over an open set, then we know how it behaves everywhere else. This seems like a gross overgeneralization, right? How can we generalize the behavior of a function over all possible numbers based on a small region? Well, that’s the beauty of complex analysis. The actual mathematical proof is too long, so we cannot discuss it here. But for now, we just need to understand that this holds true.
I don’t see the point here! How does that explain the negative fraction?
The reason we are discussing this approach is because it comes up in many places. As we can see from the figure here, this is what the zeta function looks like. There is a lot you can tell about a complex mathematical function just by looking at its behavior in some limited area. In the case of the zeta function, we have a nice definition for values of ‘s’ greater than 1. Now let’s go ahead and extend this definition to complex numbers. As it so happens, it still works if ‘s’ is a complex number as long as the real part of s is greater than 1. So we can use this information to deduce something about the zeta function. Now that we know the value of the zeta function for almost half the total area (complex numbers with real part greater than 1), we can use complex analysis to estimate its value for every other number (complex numbers with real part smaller than 1). In particular, when we substitute s=-1, we get the following:
ζ(-1) = -1/12
We can interpret this result by looking at the zeta function. Using complex analysis, we deduced that zeta function is consistently defined for every complex number. So if that were true, then we just “extrapolate” the curve to get the value of the function at s=-1. When we do that, this is the value we get when ‘s’ is -1.
This makes absolutely no sense! What kind of sorcery is this?
I understand your skepticism in accepting this result. We are going use a concept called “regularization” to understand this. This actually comes up in physics all the time. In physics, we know that there is no such thing as infinity. Everything is real and everyting is measurable. So unless there is an extreme case, nothing in nature should be infinite. So when are building systems and constructing formulations to explain the universe, nothing should lead to infinity. If it does, then it’s an indication that we are doing something wrong!
Despite all these restrictions, we still encounter calculations in physics which result in infinity. In many of these cases, the reason our calculations lead to results that are infinity is because we don’t know enough about what’s really going on. This is where regularization comes in handy. Regularization is the process by which an infinite result is replaced with a finite result in a way so that it keeps the same properties. These finite results can then be used to do calculations and make predictions, as long as the final predictions are independent of the way in which do regularization.
So can you really add up all the natural numbers and get -1/12? No! Let’s say you are formulating something and at some point, your formulation tells you to add up all the natural numbers to proceed further. It’s obvious that the result can’t be infinity, so what do we do here? The way to interpret this would be that your formulation might be secretly asking you to calculate the Riemann Zeta Function at -1, as opposed to summing up all the natural numbers. As we know from complex analysis, that value is -1/12. This formulation leads to legitimate conclusions in your system, which is what we desire. We use this formulation to explain Casimir effect, a very real thing that happens in real life.