[cohomologo]

I've never seen a physics book that treats this using the zeta function, except popular articles that try to present this calculation as mysterious. In practice, in my QFT class we needed to compute the sum

lim \epsilon -> 0+ ( \sum_{n=1}^\infty n e^{- \epsilon n} + ...),

that is the series was multiplied by a decaying exponential function with a rate of decay that goes to zero. This sum can easily be evaluated for small epsilon takes the form

sum = 1/epsilon - 1/12 + O(epsilon).

The 1/epsilon term (which goes to infinity) drops out of the final physical result when you do the calculation properly.