Does this divergent sum appear anywhere in physics?

Question

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