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

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


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


  • $\begingroup$ Reason for down-vote? $\endgroup$ – drewdles Jul 9 '16 at 10:37
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    $\begingroup$ I'd guess that you may have been downvoted because your goal in asking this question is not clear. What physics problem are you trying to work out, that having the answer to this question will help you with? $\endgroup$ – David Z Jul 9 '16 at 18:07
  • $\begingroup$ @DavidZ Well I was trying to ask if this expression appeared somewhere in physics? Perhaps it would be better if I uploaded this question on mathstackexchange? $\endgroup$ – drewdles Jul 9 '16 at 18:24
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    $\begingroup$ Well, remember that all you have so far is the opinion of one downvoter (and also one upvoter). It doesn't mean the question is off topic here. (In fact, in general votes don't indicate whether a question is on or off topic; there is only a loose correlation.) $\endgroup$ – David Z Jul 9 '16 at 18:27
  • $\begingroup$ I am confused with $k\to 1$ on the right-hand side and $k\to\infty$ on the left-hand side. You should denote one $k$ with a different letter. $\endgroup$ – Vladimir Kalitvianski Jul 9 '16 at 19:14

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