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my questiion is if this regularizatio for the Harmonic series

$$ \sum_{n=0}^{\infty}\frac{1}{(n+a)} = \frac{ -\Gamma ' (a)}{\Gamma (a)}$$

for any positive and finite 'a' appears in renormalization or in physics

what could we say about the calculation

$$ \sum_{n=0}^{\infty}\frac{log^{k}(n+a)}{(n+a)} $$ for positive and negative 'k' ?? does it appear in physics ??

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It's a good question. I don't know of any emergence, mainly because it's unusual for contributions from momenta $n$ to decrease like $1/n$ etc. (it is easier to get positive powers of $n$ from derivatives etc.). The log-based generalization is interesting but slightly less well-defined. For $a\lt 0$, there has to be a discontinuity and branching due to logarithms of negative numbers etc. –  LuboŇ° Motl Apr 11 '13 at 12:51

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