This is a whole other question that you maybe expect from me, but it is bugging me since I heard of a long time ago. Bosonic string theory (to take the simplest one of string theories) is consistent in only 26 dimensions. This critical dimension is needed to rule out things like negative propabilities, etc.
The way most physisists are calculating the 26 always boggles down to the analytic continuation for the $\zeta(s)$ function that means that $\zeta(-1)=-\tfrac{1}{12}$ and the "definition" $\zeta(-1)=\sum_{k=0}^{\infty} n$. If you don't want to use this, most physisist are relying on a technique called zeta-renormalization, which pretty much looks like the time I set down for my analysis lecture and saw "\ldots = \tfrac{\infty}{\infty}=1 on the blackboard (a leftover from a quantum field lecture that wasdn't erased).
Question: Are there mathematically sound proofs for the critical dimensions (for instance, based on some consistency rules of the inproduct on Hilbert spaces) ?
My sincere appologies for bashing theoretical physisists this way, but the leftover was really what I saw (I did even once see $\infty - \infty = 0$ in very much the same context, for that matter).