Derive the Casimir Energy in Bosonic String Theory. 

You start with the $\hat L_0$ operator and get rid  of the non-vacuum $\displaystyle\frac{\alpha_0^2}{2}+\sum_{n=1}^\infty\alpha_{-n}\cdot\alpha_n$, then you use a Ramanujam sum to do $\zeta$-function renormalisation,  from which you find out that the vacuum energy denoted by $\varepsilon_0$ is 

$$\varepsilon_0=-\frac{d-2}{24}$$       

However, the most interesting part comes when you go around [deriving](https://mathoverflow.net/a/140354/36148) the critical dimension of Bosonic String Theory. 

After which, the expression surprisingly simplifyies to a $-1$.  

For a more detailed derivation of the above stuff, see [these](http://arxiv.org/pdf/hep-th/0207142v1.pdf) lecture notes/. (Section 4) (Equation 4.5-4.10)