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I recently managed to analytically continue certain divergent series. I was hoping if anyone could tell me if they appeared somewhere in physics:

$$ \lim_{k \to \infty} \lim_{n \to \infty} \left( \sum_{r=1}^n r^{-2s+1} f( \frac{kr}{n}) \frac{k}{n} \right) = \lim_{j \to 1}\underbrace{\zeta(j) \zeta (-2s+1-j)}_{\text{removable singularity}} \int_0^\infty f(x) \, dx $$

For those interested in the derivation I uploaded it on dropbox (section $3.2$):

Derivation

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