# Divergent sum in lightcone quantization of bosonic string theory

I had the following question regarding lightcone quantization of bosonic strings - The normal ordering requirement of quantization gives us this infinite sum $\sum_{n=1}^\infty n$. This is regularized in several ways, for example by writing $$\sum_{n=1}^\infty e^{- n \epsilon } n = \frac{1}{\epsilon^2} - \frac{1}{12} + {\cal O}(\epsilon^2)$$ Most texts now simply state that the divergent part can be removed by counterterms. David Tongs notes (chap. 2 page 29) specifically state that this divergence is removed by the counterterm that restore Weyl invariance in the quantized theory (in dimensional regularization).

I would like to see this explicitly. Is there any note regarding this? Or if you have any other idea how one would systematically remove the divergence above, it would be great!

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This particular sum is also discussed here and here, and on Math.SE here. See also this Phys.SE post. –  Qmechanic Apr 3 at 17:52