Re "However, string theorists may be prone to making up their own rules.":
Seeing this bizarre result I could see it was bonkers for the reasons I've given. But the fact that physicists use the identity did make me take a second look. A website asks just this question: how did this wrong result make it into a physics textbook? I quote: "Suppose you take two conducting metallic plates and arrange them in a vacuum so that they are parallel to each other. According to classical physics, there shouldn't be any net force acting between the two plates. But classical physics doesn't reckon with the weird effects you see when you look at the world at very small scales. To do that, you need quantum physics, which tells us many very strange things. One of them is that the vacuum isn't empty, but seething with activity. So-called virtual particles pop in and out of existence all the time. This activity gives a so called zero point energy. When you try to calculate the total energy density between the two plates you get the infinite sum [a sum similar to the one we've looked at]. That’s unfortunate, because the sum diverges, which would imply an infinite energy density. That’s obviously nonsense. But what if you cheekily assume that the infinite sum equals the Riemann zeta function [don't ask!], rather than the Euler zeta function [don't ask!]? Well, then you get a finite energy density. That means there should be an attractive force between the metallic plates, which also seems ludicrous, since classical physics suggests there should be no force. But here’s the surprise. When physicists made the experiment they found that the force did exist — and it corresponded to an energy density exactly as predicted." Conclusion: bad mathematics can work if you're a quantum physicist.
